SIMPLE SOLUTIONS

REALOTHERAUXILIARY(3) - Linux man page online | Library functions

Chapter
Tue Nov 28 2017
realOTHERauxiliary(3) LAPACK realOTHERauxiliary(3)

NAME

realOTHERauxiliary

SYNOPSIS

Functions integer function ilaslc (M, N, A, LDA) ILASLC scans a matrix for its last non-zero column. integer function ilaslr (M, N, A, LDA) ILASLR scans a matrix for its last non-zero row. subroutine slabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY) SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. subroutine slacn2 (N, V, X, ISGN, EST, KASE, ISAVE) SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutine slacon (N, V, X, ISGN, EST, KASE) SLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. subroutine sladiv (A, B, C, D, P, Q) SLADIV performs complex division in real arithmetic, avoiding unnecessary overflow. subroutine sladiv1 (A, B, C, D, P, Q) real function sladiv2 (A, B, C, D, R, T) subroutine slaein (RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B, LDB, WORK, EPS3, SMLNUM, BIGNUM, INFO) SLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration. subroutine slaexc (WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK, INFO) SLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation. subroutine slag2 (A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI) SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow. subroutine slags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ) SLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel. subroutine slagtm (TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB) SLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1. subroutine slagv2 (A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, CSR, SNR) SLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular. subroutine slahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, INFO) SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. subroutine slahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY) SLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. subroutine slaic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C) SLAIC1 applies one step of incremental condition estimation. subroutine slaln2 (LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB, WR, WI, X, LDX, SCALE, XNORM, INFO) SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form. real function slangt (NORM, N, DL, D, DU) SLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix. real function slanhs (NORM, N, A, LDA, WORK) SLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix. real function slansb (NORM, UPLO, N, K, AB, LDAB, WORK) SLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix. real function slansp (NORM, UPLO, N, AP, WORK) SLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form. real function slantb (NORM, UPLO, DIAG, N, K, AB, LDAB, WORK) SLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix. real function slantp (NORM, UPLO, DIAG, N, AP, WORK) SLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form. real function slantr (NORM, UPLO, DIAG, M, N, A, LDA, WORK) SLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix. subroutine slanv2 (A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN) SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form. subroutine slapll (N, X, INCX, Y, INCY, SSMIN) SLAPLL measures the linear dependence of two vectors. subroutine slapmr (FORWRD, M, N, X, LDX, K) SLAPMR rearranges rows of a matrix as specified by a permutation vector. subroutine slapmt (FORWRD, M, N, X, LDX, K) SLAPMT performs a forward or backward permutation of the columns of a matrix. subroutine slaqp2 (M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK) SLAQP2 computes a QR factorization with column pivoting of the matrix block. subroutine slaqps (M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF) SLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3. subroutine slaqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO) SLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutine slaqr1 (N, H, LDH, SR1, SI1, SR2, SI2, V) SLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts. subroutine slaqr2 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK) SLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine slaqr3 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK) SLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). subroutine slaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO) SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. subroutine slaqr5 (WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, SR, SI, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH) SLAQR5 performs a single small-bulge multi-shift QR sweep. subroutine slaqsb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED) SLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ. subroutine slaqsp (UPLO, N, AP, S, SCOND, AMAX, EQUED) SLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ. subroutine slaqtr (LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK, INFO) SLAQTR solves a real quasi-triangular system of equations, or a complex quasi- triangular system of special form, in real arithmetic. subroutine slar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK) SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. subroutine slar2v (N, X, Y, Z, INCX, C, S, INCC) SLAR2V applies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices. subroutine slarf (SIDE, M, N, V, INCV, TAU, C, LDC, WORK) SLARF applies an elementary reflector to a general rectangular matrix. subroutine slarfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK) SLARFB applies a block reflector or its transpose to a general rectangular matrix. subroutine slarfg (N, ALPHA, X, INCX, TAU) SLARFG generates an elementary reflector (Householder matrix). subroutine slarfgp (N, ALPHA, X, INCX, TAU) SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta. subroutine slarft (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT) SLARFT forms the triangular factor T of a block reflector H = I - vtvH subroutine slarfx (SIDE, M, N, V, TAU, C, LDC, WORK) SLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10. subroutine slargv (N, X, INCX, Y, INCY, C, INCC) SLARGV generates a vector of plane rotations with real cosines and real sines. subroutine slarrv (N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MINRGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO) SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT. subroutine slartv (N, X, INCX, Y, INCY, C, S, INCC) SLARTV applies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors. subroutine slaswp (N, A, LDA, K1, K2, IPIV, INCX) SLASWP performs a series of row interchanges on a general rectangular matrix. subroutine slatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO) SLATBS solves a triangular banded system of equations. subroutine slatdf (IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV) SLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. subroutine slatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO) SLATPS solves a triangular system of equations with the matrix held in packed storage. subroutine slatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO) SLATRS solves a triangular system of equations with the scale factor set to prevent overflow. subroutine slauu2 (UPLO, N, A, LDA, INFO) SLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm). subroutine slauum (UPLO, N, A, LDA, INFO) SLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm). subroutine srscl (N, SA, SX, INCX) SRSCL multiplies a vector by the reciprocal of a real scalar. subroutine stprfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK) STPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks. Detailed Description This is the group of real other auxiliary routines Function Documentation integer function ilaslc (integer M, integer N, real, dimension( lda, * ) A, integer LDA) ILASLC scans a matrix for its last non-zero column. Purpose: ILASLC scans A for its last non-zero column. Parameters: M M is INTEGER The number of rows of the matrix A. N N is INTEGER The number of columns of the matrix A. A A is REAL array, dimension (LDA,N) The m by n matrix A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2017 integer function ilaslr (integer M, integer N, real, dimension( lda, * ) A, integer LDA) ILASLR scans a matrix for its last non-zero row. Purpose: ILASLR scans A for its last non-zero row. Parameters: M M is INTEGER The number of rows of the matrix A. N N is INTEGER The number of columns of the matrix A. A A is REAL array, dimension (LDA,N) The m by n matrix A. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slabrd (integer M, integer N, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( * ) TAUQ, real, dimension( * ) TAUP, real, dimension( ldx, * ) X, integer LDX, real, dimension( ldy, * ) Y, integer LDY) SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. Purpose: SLABRD reduces the first NB rows and columns of a real general m by n matrix A to upper or lower bidiagonal form by an orthogonal transformation Q**T * A * P, and returns the matrices X and Y which are needed to apply the transformation to the unreduced part of A. If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower bidiagonal form. This is an auxiliary routine called by SGEBRD Parameters: M M is INTEGER The number of rows in the matrix A. N N is INTEGER The number of columns in the matrix A. NB NB is INTEGER The number of leading rows and columns of A to be reduced. A A is REAL array, dimension (LDA,N) On entry, the m by n general matrix to be reduced. On exit, the first NB rows and columns of the matrix are overwritten; the rest of the array is unchanged. If m >= n, elements on and below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. If m < n, elements below the diagonal in the first NB columns, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and elements on and above the diagonal in the first NB rows, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors. See Further Details. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). D D is REAL array, dimension (NB) The diagonal elements of the first NB rows and columns of the reduced matrix. D(i) = A(i,i). E E is REAL array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix. TAUQ TAUQ is REAL array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details. TAUP TAUP is REAL array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details. X X is REAL array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A. LDX LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M). Y Y is REAL array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A. LDY LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2017 Further Details: The matrices Q and P are represented as products of elementary reflectors: Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb) Each H(i) and G(i) has the form: H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T where tauq and taup are real scalars, and v and u are real vectors. If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). The elements of the vectors v and u together form the m-by-nb matrix V and the nb-by-n matrix U**T which are needed, with X and Y, to apply the transformation to the unreduced part of the matrix, using a block update of the form: A := A - V*Y**T - X*U**T. The contents of A on exit are illustrated by the following examples with nb = 2: m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 ) ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 ) ( v1 v2 a a a ) ( v1 1 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) ( v1 v2 a a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). subroutine slacn2 (integer N, real, dimension( * ) V, real, dimension( * ) X, integer, dimension( * ) ISGN, real EST, integer KASE, integer, dimension( 3 ) ISAVE) SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. Purpose: SLACN2 estimates the 1-norm of a square, real matrix A. Reverse communication is used for evaluating matrix-vector products. Parameters: N N is INTEGER The order of the matrix. N >= 1. V V is REAL array, dimension (N) On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned). X X is REAL array, dimension (N) On an intermediate return, X should be overwritten by A * X, if KASE=1, A**T * X, if KASE=2, and SLACN2 must be re-called with all the other parameters unchanged. ISGN ISGN is INTEGER array, dimension (N) EST EST is REAL On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be unchanged from the previous call to SLACN2. On exit, EST is an estimate (a lower bound) for norm(A). KASE KASE is INTEGER On the initial call to SLACN2, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A**T * X. On the final return from SLACN2, KASE will again be 0. ISAVE ISAVE is INTEGER array, dimension (3) ISAVE is used to save variables between calls to SLACN2 Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Further Details: Originally named SONEST, dated March 16, 1988. This is a thread safe version of SLACON, which uses the array ISAVE in place of a SAVE statement, as follows: SLACON SLACN2 JUMP ISAVE(1) J ISAVE(2) ITER ISAVE(3) Contributors: Nick Higham, University of Manchester References: N.J. Higham, 'FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. subroutine slacon (integer N, real, dimension( * ) V, real, dimension( * ) X, integer, dimension( * ) ISGN, real EST, integer KASE) SLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products. Purpose: SLACON estimates the 1-norm of a square, real matrix A. Reverse communication is used for evaluating matrix-vector products. Parameters: N N is INTEGER The order of the matrix. N >= 1. V V is REAL array, dimension (N) On the final return, V = A*W, where EST = norm(V)/norm(W) (W is not returned). X X is REAL array, dimension (N) On an intermediate return, X should be overwritten by A * X, if KASE=1, A**T * X, if KASE=2, and SLACON must be re-called with all the other parameters unchanged. ISGN ISGN is INTEGER array, dimension (N) EST EST is REAL On entry with KASE = 1 or 2 and JUMP = 3, EST should be unchanged from the previous call to SLACON. On exit, EST is an estimate (a lower bound) for norm(A). KASE KASE is INTEGER On the initial call to SLACON, KASE should be 0. On an intermediate return, KASE will be 1 or 2, indicating whether X should be overwritten by A * X or A**T * X. On the final return from SLACON, KASE will again be 0. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Contributors: Nick Higham, University of Manchester. Originally named SONEST, dated March 16, 1988. References: N.J. Higham, 'FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988. subroutine sladiv (real A, real B, real C, real D, real P, real Q) SLADIV performs complex division in real arithmetic, avoiding unnecessary overflow. Purpose: SLADIV performs complex division in real arithmetic a + i*b p + i*q = --------- c + i*d The algorithm is due to Michael Baudin and Robert L. Smith and can be found in the paper "A Robust Complex Division in Scilab" Parameters: A A is REAL B B is REAL C C is REAL D D is REAL The scalars a, b, c, and d in the above expression. P P is REAL Q Q is REAL The scalars p and q in the above expression. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: January 2013 subroutine slaein (logical RIGHTV, logical NOINIT, integer N, real, dimension( ldh, * ) H, integer LDH, real WR, real WI, real, dimension( * ) VR, real, dimension( * ) VI, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) WORK, real EPS3, real SMLNUM, real BIGNUM, integer INFO) SLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration. Purpose: SLAEIN uses inverse iteration to find a right or left eigenvector corresponding to the eigenvalue (WR,WI) of a real upper Hessenberg matrix H. Parameters: RIGHTV RIGHTV is LOGICAL = .TRUE. : compute right eigenvector; = .FALSE.: compute left eigenvector. NOINIT NOINIT is LOGICAL = .TRUE. : no initial vector supplied in (VR,VI). = .FALSE.: initial vector supplied in (VR,VI). N N is INTEGER The order of the matrix H. N >= 0. H H is REAL array, dimension (LDH,N) The upper Hessenberg matrix H. LDH LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N). WR WR is REAL WI WI is REAL The real and imaginary parts of the eigenvalue of H whose corresponding right or left eigenvector is to be computed. VR VR is REAL array, dimension (N) VI VI is REAL array, dimension (N) On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain a real starting vector for inverse iteration using the real eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI must contain the real and imaginary parts of a complex starting vector for inverse iteration using the complex eigenvalue (WR,WI); otherwise VR and VI need not be set. On exit, if WI = 0.0 (real eigenvalue), VR contains the computed real eigenvector; if WI.ne.0.0 (complex eigenvalue), VR and VI contain the real and imaginary parts of the computed complex eigenvector. The eigenvector is normalized so that the component of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|. VI is not referenced if WI = 0.0. B B is REAL array, dimension (LDB,N) LDB LDB is INTEGER The leading dimension of the array B. LDB >= N+1. WORK WORK is REAL array, dimension (N) EPS3 EPS3 is REAL A small machine-dependent value which is used to perturb close eigenvalues, and to replace zero pivots. SMLNUM SMLNUM is REAL A machine-dependent value close to the underflow threshold. BIGNUM BIGNUM is REAL A machine-dependent value close to the overflow threshold. INFO INFO is INTEGER = 0: successful exit = 1: inverse iteration did not converge; VR is set to the last iterate, and so is VI if WI.ne.0.0. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slaexc (logical WANTQ, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldq, * ) Q, integer LDQ, integer J1, integer N1, integer N2, real, dimension( * ) WORK, integer INFO) SLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur canonical form, by an orthogonal similarity transformation. Purpose: SLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in an upper quasi-triangular matrix T by an orthogonal similarity transformation. T must be in Schur canonical form, that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elemnts equal and its off-diagonal elements of opposite sign. Parameters: WANTQ WANTQ is LOGICAL = .TRUE. : accumulate the transformation in the matrix Q; = .FALSE.: do not accumulate the transformation. N N is INTEGER The order of the matrix T. N >= 0. T T is REAL array, dimension (LDT,N) On entry, the upper quasi-triangular matrix T, in Schur canonical form. On exit, the updated matrix T, again in Schur canonical form. LDT LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N). Q Q is REAL array, dimension (LDQ,N) On entry, if WANTQ is .TRUE., the orthogonal matrix Q. On exit, if WANTQ is .TRUE., the updated matrix Q. If WANTQ is .FALSE., Q is not referenced. LDQ LDQ is INTEGER The leading dimension of the array Q. LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N. J1 J1 is INTEGER The index of the first row of the first block T11. N1 N1 is INTEGER The order of the first block T11. N1 = 0, 1 or 2. N2 N2 is INTEGER The order of the second block T22. N2 = 0, 1 or 2. WORK WORK is REAL array, dimension (N) INFO INFO is INTEGER = 0: successful exit = 1: the transformed matrix T would be too far from Schur form; the blocks are not swapped and T and Q are unchanged. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slag2 (real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real SAFMIN, real SCALE1, real SCALE2, real WR1, real WR2, real WI) SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow. Purpose: SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue problem A - w B, with scaling as necessary to avoid over-/underflow. The scaling factor "s" results in a modified eigenvalue equation s A - w B where s is a non-negative scaling factor chosen so that w, w B, and s A do not overflow and, if possible, do not underflow, either. Parameters: A A is REAL array, dimension (LDA, 2) On entry, the 2 x 2 matrix A. It is assumed that its 1-norm is less than 1/SAFMIN. Entries less than sqrt(SAFMIN)*norm(A) are subject to being treated as zero. LDA LDA is INTEGER The leading dimension of the array A. LDA >= 2. B B is REAL array, dimension (LDB, 2) On entry, the 2 x 2 upper triangular matrix B. It is assumed that the one-norm of B is less than 1/SAFMIN. The diagonals should be at least sqrt(SAFMIN) times the largest element of B (in absolute value); if a diagonal is smaller than that, then +/- sqrt(SAFMIN) will be used instead of that diagonal. LDB LDB is INTEGER The leading dimension of the array B. LDB >= 2. SAFMIN SAFMIN is REAL The smallest positive number s.t. 1/SAFMIN does not overflow. (This should always be SLAMCH('S') -- it is an argument in order to avoid having to call SLAMCH frequently.) SCALE1 SCALE1 is REAL A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the first eigenvalue. If the eigenvalues are complex, then the eigenvalues are ( WR1 +/- WI i ) / SCALE1 (which may lie outside the exponent range of the machine), SCALE1=SCALE2, and SCALE1 will always be positive. If the eigenvalues are real, then the first (real) eigenvalue is WR1 / SCALE1 , but this may overflow or underflow, and in fact, SCALE1 may be zero or less than the underflow threshold if the exact eigenvalue is sufficiently large. SCALE2 SCALE2 is REAL A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the second eigenvalue. If the eigenvalues are complex, then SCALE2=SCALE1. If the eigenvalues are real, then the second (real) eigenvalue is WR2 / SCALE2 , but this may overflow or underflow, and in fact, SCALE2 may be zero or less than the underflow threshold if the exact eigenvalue is sufficiently large. WR1 WR1 is REAL If the eigenvalue is real, then WR1 is SCALE1 times the eigenvalue closest to the (2,2) element of A B**(-1). If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues. WR2 WR2 is REAL If the eigenvalue is real, then WR2 is SCALE2 times the other eigenvalue. If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues. WI WI is REAL If the eigenvalue is real, then WI is zero. If the eigenvalue is complex, then WI is SCALE1 times the imaginary part of the eigenvalues. WI will always be non-negative. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2016 subroutine slags2 (logical UPPER, real A1, real A2, real A3, real B1, real B2, real B3, real CSU, real SNU, real CSV, real SNV, real CSQ, real SNQ) SLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and B such that the rows of the transformed A and B are parallel. Purpose: SLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such that if ( UPPER ) then U**T *A*Q = U**T *( A1 A2 )*Q = ( x 0 ) ( 0 A3 ) ( x x ) and V**T*B*Q = V**T *( B1 B2 )*Q = ( x 0 ) ( 0 B3 ) ( x x ) or if ( .NOT.UPPER ) then U**T *A*Q = U**T *( A1 0 )*Q = ( x x ) ( A2 A3 ) ( 0 x ) and V**T*B*Q = V**T*( B1 0 )*Q = ( x x ) ( B2 B3 ) ( 0 x ) The rows of the transformed A and B are parallel, where U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ ) ( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ ) Z**T denotes the transpose of Z. Parameters: UPPER UPPER is LOGICAL = .TRUE.: the input matrices A and B are upper triangular. = .FALSE.: the input matrices A and B are lower triangular. A1 A1 is REAL A2 A2 is REAL A3 A3 is REAL On entry, A1, A2 and A3 are elements of the input 2-by-2 upper (lower) triangular matrix A. B1 B1 is REAL B2 B2 is REAL B3 B3 is REAL On entry, B1, B2 and B3 are elements of the input 2-by-2 upper (lower) triangular matrix B. CSU CSU is REAL SNU SNU is REAL The desired orthogonal matrix U. CSV CSV is REAL SNV SNV is REAL The desired orthogonal matrix V. CSQ CSQ is REAL SNQ SNQ is REAL The desired orthogonal matrix Q. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slagtm (character TRANS, integer N, integer NRHS, real ALPHA, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU, real, dimension( ldx, * ) X, integer LDX, real BETA, real, dimension( ldb, * ) B, integer LDB) SLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1. Purpose: SLAGTM performs a matrix-vector product of the form B := alpha * A * X + beta * B where A is a tridiagonal matrix of order N, B and X are N by NRHS matrices, and alpha and beta are real scalars, each of which may be 0., 1., or -1. Parameters: TRANS TRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': No transpose, B := alpha * A * X + beta * B = 'T': Transpose, B := alpha * A'* X + beta * B = 'C': Conjugate transpose = Transpose N N is INTEGER The order of the matrix A. N >= 0. NRHS NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices X and B. ALPHA ALPHA is REAL The scalar alpha. ALPHA must be 0., 1., or -1.; otherwise, it is assumed to be 0. DL DL is REAL array, dimension (N-1) The (n-1) sub-diagonal elements of T. D D is REAL array, dimension (N) The diagonal elements of T. DU DU is REAL array, dimension (N-1) The (n-1) super-diagonal elements of T. X X is REAL array, dimension (LDX,NRHS) The N by NRHS matrix X. LDX LDX is INTEGER The leading dimension of the array X. LDX >= max(N,1). BETA BETA is REAL The scalar beta. BETA must be 0., 1., or -1.; otherwise, it is assumed to be 1. B B is REAL array, dimension (LDB,NRHS) On entry, the N by NRHS matrix B. On exit, B is overwritten by the matrix expression B := alpha * A * X + beta * B. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(N,1). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slagv2 (real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( 2 ) ALPHAR, real, dimension( 2 ) ALPHAI, real, dimension( 2 ) BETA, real CSL, real SNL, real CSR, real SNR) SLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular. Purpose: SLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B) where B is upper triangular. This routine computes orthogonal (rotation) matrices given by CSL, SNL and CSR, SNR such that 1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0 types), then [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] [ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] [ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ], 2) if the pencil (A,B) has a pair of complex conjugate eigenvalues, then [ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ] [ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ] [ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ] [ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ] where b11 >= b22 > 0. Parameters: A A is REAL array, dimension (LDA, 2) On entry, the 2 x 2 matrix A. On exit, A is overwritten by the ``A-part'' of the generalized Schur form. LDA LDA is INTEGER THe leading dimension of the array A. LDA >= 2. B B is REAL array, dimension (LDB, 2) On entry, the upper triangular 2 x 2 matrix B. On exit, B is overwritten by the ``B-part'' of the generalized Schur form. LDB LDB is INTEGER THe leading dimension of the array B. LDB >= 2. ALPHAR ALPHAR is REAL array, dimension (2) ALPHAI ALPHAI is REAL array, dimension (2) BETA BETA is REAL array, dimension (2) (ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may be zero. CSL CSL is REAL The cosine of the left rotation matrix. SNL SNL is REAL The sine of the left rotation matrix. CSR CSR is REAL The cosine of the right rotation matrix. SNR SNR is REAL The sine of the right rotation matrix. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Contributors: Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA subroutine slahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, integer INFO) SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. Purpose: SLAHQR is an auxiliary routine called by SHSEQR to update the eigenvalues and Schur decomposition already computed by SHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI. Parameters: WANTT WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required. WANTZ WANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required. N N is INTEGER The order of the matrix H. N >= 0. ILO ILO is INTEGER IHI IHI is INTEGER It is assumed that H is already upper quasi-triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). SLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but applies transformations to all of H if WANTT is .TRUE.. 1 <= ILO <= max(1,IHI); IHI <= N. H H is REAL array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO is zero and if WANTT is .TRUE., H is upper quasi-triangular in rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in standard form. If INFO is zero and WANTT is .FALSE., the contents of H are unspecified on exit. The output state of H if INFO is nonzero is given below under the description of INFO. LDH LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N). WR WR is REAL array, dimension (N) WI WI is REAL array, dimension (N) The real and imaginary parts, respectively, of the computed eigenvalues ILO to IHI are stored in the corresponding elements of WR and WI. If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i), and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i). ILOZ ILOZ is INTEGER IHIZ IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. Z Z is REAL array, dimension (LDZ,N) If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by SHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI). If WANTZ is .FALSE., Z is not referenced. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,N). INFO INFO is INTEGER = 0: successful exit .GT. 0: If INFO = i, SLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30 iterations per eigenvalue; elements i+1:ihi of WR and WI contain those eigenvalues which have been successfully computed. If INFO .GT. 0 and WANTT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ILO thorugh INFO of the final, output value of H. If INFO .GT. 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is an orthognal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regardless of the value of WANTT.) Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Further Details: 02-96 Based on modifications by David Day, Sandia National Laboratory, USA 12-04 Further modifications by Ralph Byers, University of Kansas, USA This is a modified version of SLAHQR from LAPACK version 3.0. It is (1) more robust against overflow and underflow and (2) adopts the more conservative Ahues & Tisseur stopping criterion (LAWN 122, 1997). subroutine slahr2 (integer N, integer K, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( nb ) TAU, real, dimension( ldt, nb ) T, integer LDT, real, dimension( ldy, nb ) Y, integer LDY) SLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. Purpose: SLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q**T * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T. This is an auxiliary routine called by SGEHRD. Parameters: N N is INTEGER The order of the matrix A. K K is INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. K < N. NB NB is INTEGER The number of columns to be reduced. A A is REAL array, dimension (LDA,N-K+1) On entry, the n-by-(n-k+1) general matrix A. On exit, the elements on and above the k-th subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced matrix; the elements below the k-th subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A are unchanged. See Further Details. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). TAU TAU is REAL array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. T T is REAL array, dimension (LDT,NB) The upper triangular matrix T. LDT LDT is INTEGER The leading dimension of the array T. LDT >= NB. Y Y is REAL array, dimension (LDY,NB) The n-by-nb matrix Y. LDY LDY is INTEGER The leading dimension of the array Y. LDY >= N. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Further Details: The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) . . . H(nb). Each H(i) has the form H(i) = I - tau * v * v**T where tau is a real scalar, and v is a real vector with v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i), and tau in TAU(i). The elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to apply the transformation to the unreduced part of the matrix, using an update of the form: A := (I - V*T*V**T) * (A - Y*V**T). The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: ( a a a a a ) ( a a a a a ) ( a a a a a ) ( h h a a a ) ( v1 h a a a ) ( v1 v2 a a a ) ( v1 v2 a a a ) where a denotes an element of the original matrix A, h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i). This subroutine is a slight modification of LAPACK-3.0's DLAHRD incorporating improvements proposed by Quintana-Orti and Van de Gejin. Note that the entries of A(1:K,2:NB) differ from those returned by the original LAPACK-3.0's DLAHRD routine. (This subroutine is not backward compatible with LAPACK-3.0's DLAHRD.) References: Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006. subroutine slaic1 (integer JOB, integer J, real, dimension( j ) X, real SEST, real, dimension( j ) W, real GAMMA, real SESTPR, real S, real C) SLAIC1 applies one step of incremental condition estimation. Purpose: SLAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that twonorm(L*x) = sest Then SLAIC1 computes sestpr, s, c such that the vector [ s*x ] xhat = [ c ] is an approximate singular vector of [ L 0 ] Lhat = [ w**T gamma ] in the sense that twonorm(Lhat*xhat) = sestpr. Depending on JOB, an estimate for the largest or smallest singular value is computed. Note that [s c]**T and sestpr**2 is an eigenpair of the system diag(sest*sest, 0) + [alpha gamma] * [ alpha ] [ gamma ] where alpha = x**T*w. Parameters: JOB JOB is INTEGER = 1: an estimate for the largest singular value is computed. = 2: an estimate for the smallest singular value is computed. J J is INTEGER Length of X and W X X is REAL array, dimension (J) The j-vector x. SEST SEST is REAL Estimated singular value of j by j matrix L W W is REAL array, dimension (J) The j-vector w. GAMMA GAMMA is REAL The diagonal element gamma. SESTPR SESTPR is REAL Estimated singular value of (j+1) by (j+1) matrix Lhat. S S is REAL Sine needed in forming xhat. C C is REAL Cosine needed in forming xhat. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slaln2 (logical LTRANS, integer NA, integer NW, real SMIN, real CA, real, dimension( lda, * ) A, integer LDA, real D1, real D2, real, dimension( ldb, * ) B, integer LDB, real WR, real WI, real, dimension( ldx, * ) X, integer LDX, real SCALE, real XNORM, integer INFO) SLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form. Purpose: SLALN2 solves a system of the form (ca A - w D ) X = s B or (ca A**T - w D) X = s B with possible scaling ("s") and perturbation of A. (A**T means A-transpose.) A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA real diagonal matrix, w is a real or complex value, and X and B are NA x 1 matrices -- real if w is real, complex if w is complex. NA may be 1 or 2. If w is complex, X and B are represented as NA x 2 matrices, the first column of each being the real part and the second being the imaginary part. "s" is a scaling factor (.LE. 1), computed by SLALN2, which is so chosen that X can be computed without overflow. X is further scaled if necessary to assure that norm(ca A - w D)*norm(X) is less than overflow. If both singular values of (ca A - w D) are less than SMIN, SMIN*identity will be used instead of (ca A - w D). If only one singular value is less than SMIN, one element of (ca A - w D) will be perturbed enough to make the smallest singular value roughly SMIN. If both singular values are at least SMIN, (ca A - w D) will not be perturbed. In any case, the perturbation will be at most some small multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values are computed by infinity-norm approximations, and thus will only be correct to a factor of 2 or so. Note: all input quantities are assumed to be smaller than overflow by a reasonable factor. (See BIGNUM.) Parameters: LTRANS LTRANS is LOGICAL =.TRUE.: A-transpose will be used. =.FALSE.: A will be used (not transposed.) NA NA is INTEGER The size of the matrix A. It may (only) be 1 or 2. NW NW is INTEGER 1 if "w" is real, 2 if "w" is complex. It may only be 1 or 2. SMIN SMIN is REAL The desired lower bound on the singular values of A. This should be a safe distance away from underflow or overflow, say, between (underflow/machine precision) and (machine precision * overflow ). (See BIGNUM and ULP.) CA CA is REAL The coefficient c, which A is multiplied by. A A is REAL array, dimension (LDA,NA) The NA x NA matrix A. LDA LDA is INTEGER The leading dimension of A. It must be at least NA. D1 D1 is REAL The 1,1 element in the diagonal matrix D. D2 D2 is REAL The 2,2 element in the diagonal matrix D. Not used if NA=1. B B is REAL array, dimension (LDB,NW) The NA x NW matrix B (right-hand side). If NW=2 ("w" is complex), column 1 contains the real part of B and column 2 contains the imaginary part. LDB LDB is INTEGER The leading dimension of B. It must be at least NA. WR WR is REAL The real part of the scalar "w". WI WI is REAL The imaginary part of the scalar "w". Not used if NW=1. X X is REAL array, dimension (LDX,NW) The NA x NW matrix X (unknowns), as computed by SLALN2. If NW=2 ("w" is complex), on exit, column 1 will contain the real part of X and column 2 will contain the imaginary part. LDX LDX is INTEGER The leading dimension of X. It must be at least NA. SCALE SCALE is REAL The scale factor that B must be multiplied by to insure that overflow does not occur when computing X. Thus, (ca A - w D) X will be SCALE*B, not B (ignoring perturbations of A.) It will be at most 1. XNORM XNORM is REAL The infinity-norm of X, when X is regarded as an NA x NW real matrix. INFO INFO is INTEGER An error flag. It will be set to zero if no error occurs, a negative number if an argument is in error, or a positive number if ca A - w D had to be perturbed. The possible values are: = 0: No error occurred, and (ca A - w D) did not have to be perturbed. = 1: (ca A - w D) had to be perturbed to make its smallest (or only) singular value greater than SMIN. NOTE: In the interests of speed, this routine does not check the inputs for errors. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 real function slangt (character NORM, integer N, real, dimension( * ) DL, real, dimension( * ) D, real, dimension( * ) DU) SLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix. Purpose: SLANGT returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real tridiagonal matrix A. Returns: SLANGT SLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm. Parameters: NORM NORM is CHARACTER*1 Specifies the value to be returned in SLANGT as described above. N N is INTEGER The order of the matrix A. N >= 0. When N = 0, SLANGT is set to zero. DL DL is REAL array, dimension (N-1) The (n-1) sub-diagonal elements of A. D D is REAL array, dimension (N) The diagonal elements of A. DU DU is REAL array, dimension (N-1) The (n-1) super-diagonal elements of A. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 real function slanhs (character NORM, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK) SLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix. Purpose: SLANHS returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hessenberg matrix A. Returns: SLANHS SLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm. Parameters: NORM NORM is CHARACTER*1 Specifies the value to be returned in SLANHS as described above. N N is INTEGER The order of the matrix A. N >= 0. When N = 0, SLANHS is set to zero. A A is REAL array, dimension (LDA,N) The n by n upper Hessenberg matrix A; the part of A below the first sub-diagonal is not referenced. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(N,1). WORK WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 real function slansb (character NORM, character UPLO, integer N, integer K, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK) SLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix. Purpose: SLANSB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n symmetric band matrix A, with k super-diagonals. Returns: SLANSB SLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm. Parameters: NORM NORM is CHARACTER*1 Specifies the value to be returned in SLANSB as described above. UPLO UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the band matrix A is supplied. = 'U': Upper triangular part is supplied = 'L': Lower triangular part is supplied N N is INTEGER The order of the matrix A. N >= 0. When N = 0, SLANSB is set to zero. K K is INTEGER The number of super-diagonals or sub-diagonals of the band matrix A. K >= 0. AB AB is REAL array, dimension (LDAB,N) The upper or lower triangle of the symmetric band matrix A, stored in the first K+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k). LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1. WORK WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 real function slansp (character NORM, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) WORK) SLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form. Purpose: SLANSP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix A, supplied in packed form. Returns: SLANSP SLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm. Parameters: NORM NORM is CHARACTER*1 Specifies the value to be returned in SLANSP as described above. UPLO UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is supplied. = 'U': Upper triangular part of A is supplied = 'L': Lower triangular part of A is supplied N N is INTEGER The order of the matrix A. N >= 0. When N = 0, SLANSP is set to zero. AP AP is REAL array, dimension (N*(N+1)/2) The upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. WORK WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 real function slantb (character NORM, character UPLO, character DIAG, integer N, integer K, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK) SLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix. Purpose: SLANTB returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of an n by n triangular band matrix A, with ( k + 1 ) diagonals. Returns: SLANTB SLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm. Parameters: NORM NORM is CHARACTER*1 Specifies the value to be returned in SLANTB as described above. UPLO UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular DIAG DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular N N is INTEGER The order of the matrix A. N >= 0. When N = 0, SLANTB is set to zero. K K is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals of the matrix A if UPLO = 'L'. K >= 0. AB AB is REAL array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first k+1 rows of AB. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k). Note that when DIAG = 'U', the elements of the array AB corresponding to the diagonal elements of the matrix A are not referenced, but are assumed to be one. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= K+1. WORK WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 real function slantp (character NORM, character UPLO, character DIAG, integer N, real, dimension( * ) AP, real, dimension( * ) WORK) SLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form. Purpose: SLANTP returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix A, supplied in packed form. Returns: SLANTP SLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm. Parameters: NORM NORM is CHARACTER*1 Specifies the value to be returned in SLANTP as described above. UPLO UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular DIAG DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular N N is INTEGER The order of the matrix A. N >= 0. When N = 0, SLANTP is set to zero. AP AP is REAL array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. Note that when DIAG = 'U', the elements of the array AP corresponding to the diagonal elements of the matrix A are not referenced, but are assumed to be one. WORK WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= N when NORM = 'I'; otherwise, WORK is not referenced. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 real function slantr (character NORM, character UPLO, character DIAG, integer M, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK) SLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix. Purpose: SLANTR returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A. Returns: SLANTR SLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm' ( ( norm1(A), NORM = '1', 'O' or 'o' ( ( normI(A), NORM = 'I' or 'i' ( ( normF(A), NORM = 'F', 'f', 'E' or 'e' where norm1 denotes the one norm of a matrix (maximum column sum), normI denotes the infinity norm of a matrix (maximum row sum) and normF denotes the Frobenius norm of a matrix (square root of sum of squares). Note that max(abs(A(i,j))) is not a consistent matrix norm. Parameters: NORM NORM is CHARACTER*1 Specifies the value to be returned in SLANTR as described above. UPLO UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower trapezoidal. = 'U': Upper trapezoidal = 'L': Lower trapezoidal Note that A is triangular instead of trapezoidal if M = N. DIAG DIAG is CHARACTER*1 Specifies whether or not the matrix A has unit diagonal. = 'N': Non-unit diagonal = 'U': Unit diagonal M M is INTEGER The number of rows of the matrix A. M >= 0, and if UPLO = 'U', M <= N. When M = 0, SLANTR is set to zero. N N is INTEGER The number of columns of the matrix A. N >= 0, and if UPLO = 'L', N <= M. When N = 0, SLANTR is set to zero. A A is REAL array, dimension (LDA,N) The trapezoidal matrix A (A is triangular if M = N). If UPLO = 'U', the leading m by n upper trapezoidal part of the array A contains the upper trapezoidal matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading m by n lower trapezoidal part of the array A contains the lower trapezoidal matrix, and the strictly upper triangular part of A is not referenced. Note that when DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be one. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(M,1). WORK WORK is REAL array, dimension (MAX(1,LWORK)), where LWORK >= M when NORM = 'I'; otherwise, WORK is not referenced. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slanv2 (real A, real B, real C, real D, real RT1R, real RT1I, real RT2R, real RT2I, real CS, real SN) SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form. Purpose: SLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard form: [ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ] [ C D ] [ SN CS ] [ CC DD ] [-SN CS ] where either 1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or 2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex conjugate eigenvalues. Parameters: A A is REAL B B is REAL C C is REAL D D is REAL On entry, the elements of the input matrix. On exit, they are overwritten by the elements of the standardised Schur form. RT1R RT1R is REAL RT1I RT1I is REAL RT2R RT2R is REAL RT2I RT2I is REAL The real and imaginary parts of the eigenvalues. If the eigenvalues are a complex conjugate pair, RT1I > 0. CS CS is REAL SN SN is REAL Parameters of the rotation matrix. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Further Details: Modified by V. Sima, Research Institute for Informatics, Bucharest, Romania, to reduce the risk of cancellation errors, when computing real eigenvalues, and to ensure, if possible, that abs(RT1R) >= abs(RT2R). subroutine slapll (integer N, real, dimension( * ) X, integer INCX, real, dimension( * ) Y, integer INCY, real SSMIN) SLAPLL measures the linear dependence of two vectors. Purpose: Given two column vectors X and Y, let A = ( X Y ). The subroutine first computes the QR factorization of A = Q*R, and then computes the SVD of the 2-by-2 upper triangular matrix R. The smaller singular value of R is returned in SSMIN, which is used as the measurement of the linear dependency of the vectors X and Y. Parameters: N N is INTEGER The length of the vectors X and Y. X X is REAL array, dimension (1+(N-1)*INCX) On entry, X contains the N-vector X. On exit, X is overwritten. INCX INCX is INTEGER The increment between successive elements of X. INCX > 0. Y Y is REAL array, dimension (1+(N-1)*INCY) On entry, Y contains the N-vector Y. On exit, Y is overwritten. INCY INCY is INTEGER The increment between successive elements of Y. INCY > 0. SSMIN SSMIN is REAL The smallest singular value of the N-by-2 matrix A = ( X Y ). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slapmr (logical FORWRD, integer M, integer N, real, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K) SLAPMR rearranges rows of a matrix as specified by a permutation vector. Purpose: SLAPMR rearranges the rows of the M by N matrix X as specified by the permutation K(1),K(2),...,K(M) of the integers 1,...,M. If FORWRD = .TRUE., forward permutation: X(K(I),*) is moved X(I,*) for I = 1,2,...,M. If FORWRD = .FALSE., backward permutation: X(I,*) is moved to X(K(I),*) for I = 1,2,...,M. Parameters: FORWRD FORWRD is LOGICAL = .TRUE., forward permutation = .FALSE., backward permutation M M is INTEGER The number of rows of the matrix X. M >= 0. N N is INTEGER The number of columns of the matrix X. N >= 0. X X is REAL array, dimension (LDX,N) On entry, the M by N matrix X. On exit, X contains the permuted matrix X. LDX LDX is INTEGER The leading dimension of the array X, LDX >= MAX(1,M). K K is INTEGER array, dimension (M) On entry, K contains the permutation vector. K is used as internal workspace, but reset to its original value on output. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slapmt (logical FORWRD, integer M, integer N, real, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K) SLAPMT performs a forward or backward permutation of the columns of a matrix. Purpose: SLAPMT rearranges the columns of the M by N matrix X as specified by the permutation K(1),K(2),...,K(N) of the integers 1,...,N. If FORWRD = .TRUE., forward permutation: X(*,K(J)) is moved X(*,J) for J = 1,2,...,N. If FORWRD = .FALSE., backward permutation: X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N. Parameters: FORWRD FORWRD is LOGICAL = .TRUE., forward permutation = .FALSE., backward permutation M M is INTEGER The number of rows of the matrix X. M >= 0. N N is INTEGER The number of columns of the matrix X. N >= 0. X X is REAL array, dimension (LDX,N) On entry, the M by N matrix X. On exit, X contains the permuted matrix X. LDX LDX is INTEGER The leading dimension of the array X, LDX >= MAX(1,M). K K is INTEGER array, dimension (N) On entry, K contains the permutation vector. K is used as internal workspace, but reset to its original value on output. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slaqp2 (integer M, integer N, integer OFFSET, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU, real, dimension( * ) VN1, real, dimension( * ) VN2, real, dimension( * ) WORK) SLAQP2 computes a QR factorization with column pivoting of the matrix block. Purpose: SLAQP2 computes a QR factorization with column pivoting of the block A(OFFSET+1:M,1:N). The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. Parameters: M M is INTEGER The number of rows of the matrix A. M >= 0. N N is INTEGER The number of columns of the matrix A. N >= 0. OFFSET OFFSET is INTEGER The number of rows of the matrix A that must be pivoted but no factorized. OFFSET >= 0. A A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the upper triangle of block A(OFFSET+1:M,1:N) is the triangular factor obtained; the elements in block A(OFFSET+1:M,1:N) below the diagonal, together with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. Block A(1:OFFSET,1:N) has been accordingly pivoted, but no factorized. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). JPVT JPVT is INTEGER array, dimension (N) On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted to the front of A*P (a leading column); if JPVT(i) = 0, the i-th column of A is a free column. On exit, if JPVT(i) = k, then the i-th column of A*P was the k-th column of A. TAU TAU is REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors. VN1 VN1 is REAL array, dimension (N) The vector with the partial column norms. VN2 VN2 is REAL array, dimension (N) The vector with the exact column norms. WORK WORK is REAL array, dimension (N) Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Contributors: G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia. References: LAPACK Working Note 176 subroutine slaqps (integer M, integer N, integer OFFSET, integer NB, integer KB, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU, real, dimension( * ) VN1, real, dimension( * ) VN2, real, dimension( * ) AUXV, real, dimension( ldf, * ) F, integer LDF) SLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3. Purpose: SLAQPS computes a step of QR factorization with column pivoting of a real M-by-N matrix A by using Blas-3. It tries to factorize NB columns from A starting from the row OFFSET+1, and updates all of the matrix with Blas-3 xGEMM. In some cases, due to catastrophic cancellations, it cannot factorize NB columns. Hence, the actual number of factorized columns is returned in KB. Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized. Parameters: M M is INTEGER The number of rows of the matrix A. M >= 0. N N is INTEGER The number of columns of the matrix A. N >= 0 OFFSET OFFSET is INTEGER The number of rows of A that have been factorized in previous steps. NB NB is INTEGER The number of columns to factorize. KB KB is INTEGER The number of columns actually factorized. A A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, block A(OFFSET+1:M,1:KB) is the triangular factor obtained and block A(1:OFFSET,1:N) has been accordingly pivoted, but no factorized. The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has been updated. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). JPVT JPVT is INTEGER array, dimension (N) JPVT(I) = K <==> Column K of the full matrix A has been permuted into position I in AP. TAU TAU is REAL array, dimension (KB) The scalar factors of the elementary reflectors. VN1 VN1 is REAL array, dimension (N) The vector with the partial column norms. VN2 VN2 is REAL array, dimension (N) The vector with the exact column norms. AUXV AUXV is REAL array, dimension (NB) Auxiliar vector. F F is REAL array, dimension (LDF,NB) Matrix F**T = L*Y**T*A. LDF LDF is INTEGER The leading dimension of the array F. LDF >= max(1,N). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Contributors: G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia. References: LAPACK Working Note 176 subroutine slaqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO) SLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. Purpose: SLAQR0 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. Optionally Z may be postmultiplied into an input orthogonal matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. Parameters: WANTT WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required. WANTZ WANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required. N N is INTEGER The order of the matrix H. N .GE. 0. ILO ILO is INTEGER IHI IHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1, H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to SGEBAL, and then passed to SGEHRD when the matrix output by SGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0. H H is REAL array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO.GT.0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. LDH LDH is INTEGER The leading dimension of the array H. LDH .GE. max(1,N). WR WR is REAL array, dimension (IHI) WI WI is REAL array, dimension (IHI) The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI). If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i). ILOZ ILOZ is INTEGER IHIZ IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N. Z Z is REAL array, dimension (LDZ,IHI) If WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO.GT.0 is given under the description of INFO below.) LDZ LDZ is INTEGER The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1. WORK WORK is REAL array, dimension LWORK On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = -1, then SLAQR0 does a workspace query. In this case, SLAQR0 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed. INFO INFO is INTEGER = 0: successful exit .GT. 0: if INFO = i, SLAQR0 failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 0 and WANT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigen- values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO .GT. 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is an orthogonal matrix. The final value of H is upper Hessenberg and quasi-triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal matrix in (*) (regard- less of the value of WANTT.) If INFO .GT. 0 and WANTZ is .FALSE., then Z is not accessed. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Contributors: Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA References: K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002. subroutine slaqr1 (integer N, real, dimension( ldh, * ) H, integer LDH, real SR1, real SI1, real SR2, real SI2, real, dimension( * ) V) SLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts. Purpose: Given a 2-by-2 or 3-by-3 matrix H, SLAQR1 sets v to a scalar multiple of the first column of the product (*) K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I) scaling to avoid overflows and most underflows. It is assumed that either 1) sr1 = sr2 and si1 = -si2 or 2) si1 = si2 = 0. This is useful for starting double implicit shift bulges in the QR algorithm. Parameters: N N is INTEGER Order of the matrix H. N must be either 2 or 3. H H is REAL array, dimension (LDH,N) The 2-by-2 or 3-by-3 matrix H in (*). LDH LDH is INTEGER The leading dimension of H as declared in the calling procedure. LDH.GE.N SR1 SR1 is REAL SI1 SI1 is REAL SR2 SR2 is REAL SI2 SI2 is REAL The shifts in (*). V V is REAL array, dimension (N) A scalar multiple of the first column of the matrix K in (*). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2017 Contributors: Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA subroutine slaqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, real, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, real, dimension( * ) SR, real, dimension( * ) SI, real, dimension( ldv, * ) V, integer LDV, integer NH, real, dimension( ldt, * ) T, integer LDT, integer NV, real, dimension( ldwv, * ) WV, integer LDWV, real, dimension( * ) WORK, integer LWORK) SLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). Purpose: SLAQR2 is identical to SLAQR3 except that it avoids recursion by calling SLAHQR instead of SLAQR4. Aggressive early deflation: This subroutine accepts as input an upper Hessenberg matrix H and performs an orthogonal similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been over- written by a new Hessenberg matrix that is a perturbation of an orthogonal similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries. Parameters: WANTT WANTT is LOGICAL If .TRUE., then the Hessenberg matrix H is fully updated so that the quasi-triangular Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then only enough of H is updated to preserve the eigenvalues. WANTZ WANTZ is LOGICAL If .TRUE., then the orthogonal matrix Z is updated so so that the orthogonal Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced. N N is INTEGER The order of the matrix H and (if WANTZ is .TRUE.) the order of the orthogonal matrix Z. KTOP KTOP is INTEGER It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix. KBOT KBOT is INTEGER It is assumed without a check that either KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix. NW NW is INTEGER Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1). H H is REAL array, dimension (LDH,N) On input the initial N-by-N section of H stores the Hessenberg matrix undergoing aggressive early deflation. On output H has been transformed by an orthogonal similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries. LDH LDH is INTEGER Leading dimension of H just as declared in the calling subroutine. N .LE. LDH ILOZ ILOZ is INTEGER IHIZ IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N. Z Z is REAL array, dimension (LDZ,N) IF WANTZ is .TRUE., then on output, the orthogonal similarity transformation mentioned above has been accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ is .FALSE., then Z is unreferenced. LDZ LDZ is INTEGER The leading dimension of Z just as declared in the calling subroutine. 1 .LE. LDZ. NS NS is INTEGER The number of unconverged (ie approximate) eigenvalues returned in SR and SI that may be used as shifts by the calling subroutine. ND ND is INTEGER The number of converged eigenvalues uncovered by this subroutine. SR SR is REAL array, dimension (KBOT) SI SI is REAL array, dimension (KBOT) On output, the real and imaginary parts of approximate eigenvalues that may be used for shifts are stored in SR(KBOT-ND-NS+1) through SR(KBOT-ND) and SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively. The real and imaginary parts of converged eigenvalues are stored in SR(KBOT-ND+1) through SR(KBOT) and SI(KBOT-ND+1) through SI(KBOT), respectively. V V is REAL array, dimension (LDV,NW) An NW-by-NW work array. LDV LDV is INTEGER The leading dimension of V just as declared in the calling subroutine. NW .LE. LDV NH NH is INTEGER The number of columns of T. NH.GE.NW. T T is REAL array, dimension (LDT,NW) LDT LDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW .LE. LDT NV NV is INTEGER The number of rows of work array WV available for workspace. NV.GE.NW. WV WV is REAL array, dimension (LDWV,NW) LDWV LDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW .LE. LDV WORK WORK is REAL array, dimension (LWORK) On exit, WORK(1) is set to an estimate of the optimal value of LWORK for the given values of N, NW, KTOP and KBOT. LWORK LWORK is INTEGER The dimension of the work array WORK. LWORK = 2*NW suffices, but greater efficiency may result from larger values of LWORK. If LWORK = -1, then a workspace query is assumed; SLAQR2 only estimates the optimal workspace size for the given values of N, NW, KTOP and KBOT. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2017 Contributors: Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA subroutine slaqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, real, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, real, dimension( * ) SR, real, dimension( * ) SI, real, dimension( ldv, * ) V, integer LDV, integer NH, real, dimension( ldt, * ) T, integer LDT, integer NV, real, dimension( ldwv, * ) WV, integer LDWV, real, dimension( * ) WORK, integer LWORK) SLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). Purpose: Aggressive early deflation: SLAQR3 accepts as input an upper Hessenberg matrix H and performs an orthogonal similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been over- written by a new Hessenberg matrix that is a perturbation of an orthogonal similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries. Parameters: WANTT WANTT is LOGICAL If .TRUE., then the Hessenberg matrix H is fully updated so that the quasi-triangular Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then only enough of H is updated to preserve the eigenvalues. WANTZ WANTZ is LOGICAL If .TRUE., then the orthogonal matrix Z is updated so so that the orthogonal Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced. N N is INTEGER The order of the matrix H and (if WANTZ is .TRUE.) the order of the orthogonal matrix Z. KTOP KTOP is INTEGER It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix. KBOT KBOT is INTEGER It is assumed without a check that either KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix. NW NW is INTEGER Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1). H H is REAL array, dimension (LDH,N) On input the initial N-by-N section of H stores the Hessenberg matrix undergoing aggressive early deflation. On output H has been transformed by an orthogonal similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries. LDH LDH is INTEGER Leading dimension of H just as declared in the calling subroutine. N .LE. LDH ILOZ ILOZ is INTEGER IHIZ IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N. Z Z is REAL array, dimension (LDZ,N) IF WANTZ is .TRUE., then on output, the orthogonal similarity transformation mentioned above has been accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ is .FALSE., then Z is unreferenced. LDZ LDZ is INTEGER The leading dimension of Z just as declared in the calling subroutine. 1 .LE. LDZ. NS NS is INTEGER The number of unconverged (ie approximate) eigenvalues returned in SR and SI that may be used as shifts by the calling subroutine. ND ND is INTEGER The number of converged eigenvalues uncovered by this subroutine. SR SR is REAL array, dimension (KBOT) SI SI is REAL array, dimension (KBOT) On output, the real and imaginary parts of approximate eigenvalues that may be used for shifts are stored in SR(KBOT-ND-NS+1) through SR(KBOT-ND) and SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively. The real and imaginary parts of converged eigenvalues are stored in SR(KBOT-ND+1) through SR(KBOT) and SI(KBOT-ND+1) through SI(KBOT), respectively. V V is REAL array, dimension (LDV,NW) An NW-by-NW work array. LDV LDV is INTEGER The leading dimension of V just as declared in the calling subroutine. NW .LE. LDV NH NH is INTEGER The number of columns of T. NH.GE.NW. T T is REAL array, dimension (LDT,NW) LDT LDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW .LE. LDT NV NV is INTEGER The number of rows of work array WV available for workspace. NV.GE.NW. WV WV is REAL array, dimension (LDWV,NW) LDWV LDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW .LE. LDV WORK WORK is REAL array, dimension (LWORK) On exit, WORK(1) is set to an estimate of the optimal value of LWORK for the given values of N, NW, KTOP and KBOT. LWORK LWORK is INTEGER The dimension of the work array WORK. LWORK = 2*NW suffices, but greater efficiency may result from larger values of LWORK. If LWORK = -1, then a workspace query is assumed; SLAQR3 only estimates the optimal workspace size for the given values of N, NW, KTOP and KBOT. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2016 Contributors: Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA subroutine slaqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO) SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. Purpose: SLAQR4 implements one level of recursion for SLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by SLAQR0 and, for large enough deflation window size, it may be called by SLAQR3. This subroutine is identical to SLAQR0 except that it calls SLAQR2 instead of SLAQR3. SLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. Optionally Z may be postmultiplied into an input orthogonal matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T. Parameters: WANTT WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required. WANTZ WANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required. N N is INTEGER The order of the matrix H. N .GE. 0. ILO ILO is INTEGER IHI IHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1, H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to SGEBAL, and then passed to SGEHRD when the matrix output by SGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0. H H is REAL array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO.GT.0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. LDH LDH is INTEGER The leading dimension of the array H. LDH .GE. max(1,N). WR WR is REAL array, dimension (IHI) WI WI is REAL array, dimension (IHI) The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI). If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i). ILOZ ILOZ is INTEGER IHIZ IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N. Z Z is REAL array, dimension (LDZ,IHI) If WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO.GT.0 is given under the description of INFO below.) LDZ LDZ is INTEGER The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1. WORK WORK is REAL array, dimension LWORK On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK. LWORK LWORK is INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = -1, then SLAQR4 does a workspace query. In this case, SLAQR4 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed. INFO INFO is INTEGER batim INFO is INTEGER = 0: successful exit .GT. 0: if INFO = i, SLAQR4 failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 0 and WANT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigen- values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO .GT. 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is a orthogonal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal matrix in (*) (regard- less of the value of WANTT.) If INFO .GT. 0 and WANTZ is .FALSE., then Z is not accessed. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Contributors: Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA References: K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002. subroutine slaqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP, integer KBOT, integer NSHFTS, real, dimension( * ) SR, real, dimension( * ) SI, real, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldu, * ) U, integer LDU, integer NV, real, dimension( ldwv, * ) WV, integer LDWV, integer NH, real, dimension( ldwh, * ) WH, integer LDWH) SLAQR5 performs a single small-bulge multi-shift QR sweep. Purpose: SLAQR5, called by SLAQR0, performs a single small-bulge multi-shift QR sweep. Parameters: WANTT WANTT is LOGICAL WANTT = .true. if the quasi-triangular Schur factor is being computed. WANTT is set to .false. otherwise. WANTZ WANTZ is LOGICAL WANTZ = .true. if the orthogonal Schur factor is being computed. WANTZ is set to .false. otherwise. KACC22 KACC22 is INTEGER with value 0, 1, or 2. Specifies the computation mode of far-from-diagonal orthogonal updates. = 0: SLAQR5 does not accumulate reflections and does not use matrix-matrix multiply to update far-from-diagonal matrix entries. = 1: SLAQR5 accumulates reflections and uses matrix-matrix multiply to update the far-from-diagonal matrix entries. = 2: SLAQR5 accumulates reflections, uses matrix-matrix multiply to update the far-from-diagonal matrix entries, and takes advantage of 2-by-2 block structure during matrix multiplies. N N is INTEGER N is the order of the Hessenberg matrix H upon which this subroutine operates. KTOP KTOP is INTEGER KBOT KBOT is INTEGER These are the first and last rows and columns of an isolated diagonal block upon which the QR sweep is to be applied. It is assumed without a check that either KTOP = 1 or H(KTOP,KTOP-1) = 0 and either KBOT = N or H(KBOT+1,KBOT) = 0. NSHFTS NSHFTS is INTEGER NSHFTS gives the number of simultaneous shifts. NSHFTS must be positive and even. SR SR is REAL array, dimension (NSHFTS) SI SI is REAL array, dimension (NSHFTS) SR contains the real parts and SI contains the imaginary parts of the NSHFTS shifts of origin that define the multi-shift QR sweep. On output SR and SI may be reordered. H H is REAL array, dimension (LDH,N) On input H contains a Hessenberg matrix. On output a multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied to the isolated diagonal block in rows and columns KTOP through KBOT. LDH LDH is INTEGER LDH is the leading dimension of H just as declared in the calling procedure. LDH.GE.MAX(1,N). ILOZ ILOZ is INTEGER IHIZ IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. IHIZ .LE. N Z Z is REAL array, dimension (LDZ,IHIZ) If WANTZ = .TRUE., then the QR Sweep orthogonal similarity transformation is accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ = .FALSE., then Z is unreferenced. LDZ LDZ is INTEGER LDA is the leading dimension of Z just as declared in the calling procedure. LDZ.GE.N. V V is REAL array, dimension (LDV,NSHFTS/2) LDV LDV is INTEGER LDV is the leading dimension of V as declared in the calling procedure. LDV.GE.3. U U is REAL array, dimension (LDU,3*NSHFTS-3) LDU LDU is INTEGER LDU is the leading dimension of U just as declared in the in the calling subroutine. LDU.GE.3*NSHFTS-3. NH NH is INTEGER NH is the number of columns in array WH available for workspace. NH.GE.1. WH WH is REAL array, dimension (LDWH,NH) LDWH LDWH is INTEGER Leading dimension of WH just as declared in the calling procedure. LDWH.GE.3*NSHFTS-3. NV NV is INTEGER NV is the number of rows in WV agailable for workspace. NV.GE.1. WV WV is REAL array, dimension (LDWV,3*NSHFTS-3) LDWV LDWV is INTEGER LDWV is the leading dimension of WV as declared in the in the calling subroutine. LDWV.GE.NV. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2016 Contributors: Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA References: K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002. subroutine slaqsb (character UPLO, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, character EQUED) SLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ. Purpose: SLAQSB equilibrates a symmetric band matrix A using the scaling factors in the vector S. Parameters: UPLO UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals if UPLO = 'L'. KD >= 0. AB AB is REAL array, dimension (LDAB,N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, if INFO = 0, the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T of the band matrix A, in the same storage format as A. LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. S S is REAL array, dimension (N) The scale factors for A. SCOND SCOND is REAL Ratio of the smallest S(i) to the largest S(i). AMAX AMAX is REAL Absolute value of largest matrix entry. EQUED EQUED is CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). Internal Parameters: THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slaqsp (character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) S, real SCOND, real AMAX, character EQUED) SLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ. Purpose: SLAQSP equilibrates a symmetric matrix A using the scaling factors in the vector S. Parameters: UPLO UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored. = 'U': Upper triangular = 'L': Lower triangular N N is INTEGER The order of the matrix A. N >= 0. AP AP is REAL array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. On exit, the equilibrated matrix: diag(S) * A * diag(S), in the same storage format as A. S S is REAL array, dimension (N) The scale factors for A. SCOND SCOND is REAL Ratio of the smallest S(i) to the largest S(i). AMAX AMAX is REAL Absolute value of largest matrix entry. EQUED EQUED is CHARACTER*1 Specifies whether or not equilibration was done. = 'N': No equilibration. = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). Internal Parameters: THRESH is a threshold value used to decide if scaling should be done based on the ratio of the scaling factors. If SCOND < THRESH, scaling is done. LARGE and SMALL are threshold values used to decide if scaling should be done based on the absolute size of the largest matrix element. If AMAX > LARGE or AMAX < SMALL, scaling is done. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slaqtr (logical LTRAN, logical LREAL, integer N, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) B, real W, real SCALE, real, dimension( * ) X, real, dimension( * ) WORK, integer INFO) SLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular system of special form, in real arithmetic. Purpose: SLAQTR solves the real quasi-triangular system op(T)*p = scale*c, if LREAL = .TRUE. or the complex quasi-triangular systems op(T + iB)*(p+iq) = scale*(c+id), if LREAL = .FALSE. in real arithmetic, where T is upper quasi-triangular. If LREAL = .FALSE., then the first diagonal block of T must be 1 by 1, B is the specially structured matrix B = [ b(1) b(2) ... b(n) ] [ w ] [ w ] [ . ] [ w ] op(A) = A or A**T, A**T denotes the transpose of matrix A. On input, X = [ c ]. On output, X = [ p ]. [ d ] [ q ] This subroutine is designed for the condition number estimation in routine STRSNA. Parameters: LTRAN LTRAN is LOGICAL On entry, LTRAN specifies the option of conjugate transpose: = .FALSE., op(T+i*B) = T+i*B, = .TRUE., op(T+i*B) = (T+i*B)**T. LREAL LREAL is LOGICAL On entry, LREAL specifies the input matrix structure: = .FALSE., the input is complex = .TRUE., the input is real N N is INTEGER On entry, N specifies the order of T+i*B. N >= 0. T T is REAL array, dimension (LDT,N) On entry, T contains a matrix in Schur canonical form. If LREAL = .FALSE., then the first diagonal block of T must be 1 by 1. LDT LDT is INTEGER The leading dimension of the matrix T. LDT >= max(1,N). B B is REAL array, dimension (N) On entry, B contains the elements to form the matrix B as described above. If LREAL = .TRUE., B is not referenced. W W is REAL On entry, W is the diagonal element of the matrix B. If LREAL = .TRUE., W is not referenced. SCALE SCALE is REAL On exit, SCALE is the scale factor. X X is REAL array, dimension (2*N) On entry, X contains the right hand side of the system. On exit, X is overwritten by the solution. WORK WORK is REAL array, dimension (N) INFO INFO is INTEGER On exit, INFO is set to 0: successful exit. 1: the some diagonal 1 by 1 block has been perturbed by a small number SMIN to keep nonsingularity. 2: the some diagonal 2 by 2 block has been perturbed by a small number in SLALN2 to keep nonsingularity. NOTE: In the interests of speed, this routine does not check the inputs for errors. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slar1v (integer N, integer B1, integer BN, real LAMBDA, real, dimension( * ) D, real, dimension( * ) L, real, dimension( * ) LD, real, dimension( * ) LLD, real PIVMIN, real GAPTOL, real, dimension( * ) Z, logical WANTNC, integer NEGCNT, real ZTZ, real MINGMA, integer R, integer, dimension( * ) ISUPPZ, real NRMINV, real RESID, real RQCORR, real, dimension( * ) WORK) SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI. Purpose: SLAR1V computes the (scaled) r-th column of the inverse of the sumbmatrix in rows B1 through BN of the tridiagonal matrix L D L**T - sigma I. When sigma is close to an eigenvalue, the computed vector is an accurate eigenvector. Usually, r corresponds to the index where the eigenvector is largest in magnitude. The following steps accomplish this computation : (a) Stationary qd transform, L D L**T - sigma I = L(+) D(+) L(+)**T, (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T, (c) Computation of the diagonal elements of the inverse of L D L**T - sigma I by combining the above transforms, and choosing r as the index where the diagonal of the inverse is (one of the) largest in magnitude. (d) Computation of the (scaled) r-th column of the inverse using the twisted factorization obtained by combining the top part of the the stationary and the bottom part of the progressive transform. Parameters: N N is INTEGER The order of the matrix L D L**T. B1 B1 is INTEGER First index of the submatrix of L D L**T. BN BN is INTEGER Last index of the submatrix of L D L**T. LAMBDA LAMBDA is REAL The shift. In order to compute an accurate eigenvector, LAMBDA should be a good approximation to an eigenvalue of L D L**T. L L is REAL array, dimension (N-1) The (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements 1 to N-1. D D is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D. LD LD is REAL array, dimension (N-1) The n-1 elements L(i)*D(i). LLD LLD is REAL array, dimension (N-1) The n-1 elements L(i)*L(i)*D(i). PIVMIN PIVMIN is REAL The minimum pivot in the Sturm sequence. GAPTOL GAPTOL is REAL Tolerance that indicates when eigenvector entries are negligible w.r.t. their contribution to the residual. Z Z is REAL array, dimension (N) On input, all entries of Z must be set to 0. On output, Z contains the (scaled) r-th column of the inverse. The scaling is such that Z(R) equals 1. WANTNC WANTNC is LOGICAL Specifies whether NEGCNT has to be computed. NEGCNT NEGCNT is INTEGER If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin in the matrix factorization L D L**T, and NEGCNT = -1 otherwise. ZTZ ZTZ is REAL The square of the 2-norm of Z. MINGMA MINGMA is REAL The reciprocal of the largest (in magnitude) diagonal element of the inverse of L D L**T - sigma I. R R is INTEGER The twist index for the twisted factorization used to compute Z. On input, 0 <= R <= N. If R is input as 0, R is set to the index where (L D L**T - sigma I)^{-1} is largest in magnitude. If 1 <= R <= N, R is unchanged. On output, R contains the twist index used to compute Z. Ideally, R designates the position of the maximum entry in the eigenvector. ISUPPZ ISUPPZ is INTEGER array, dimension (2) The support of the vector in Z, i.e., the vector Z is nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ). NRMINV NRMINV is REAL NRMINV = 1/SQRT( ZTZ ) RESID RESID is REAL The residual of the FP vector. RESID = ABS( MINGMA )/SQRT( ZTZ ) RQCORR RQCORR is REAL The Rayleigh Quotient correction to LAMBDA. RQCORR = MINGMA*TMP WORK WORK is REAL array, dimension (4*N) Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Contributors: Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USA subroutine slar2v (integer N, real, dimension( * ) X, real, dimension( * ) Y, real, dimension( * ) Z, integer INCX, real, dimension( * ) C, real, dimension( * ) S, integer INCC) SLAR2V applies a vector of plane rotations with real cosines and real sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices. Purpose: SLAR2V applies a vector of real plane rotations from both sides to a sequence of 2-by-2 real symmetric matrices, defined by the elements of the vectors x, y and z. For i = 1,2,...,n ( x(i) z(i) ) := ( c(i) s(i) ) ( x(i) z(i) ) ( c(i) -s(i) ) ( z(i) y(i) ) ( -s(i) c(i) ) ( z(i) y(i) ) ( s(i) c(i) ) Parameters: N N is INTEGER The number of plane rotations to be applied. X X is REAL array, dimension (1+(N-1)*INCX) The vector x. Y Y is REAL array, dimension (1+(N-1)*INCX) The vector y. Z Z is REAL array, dimension (1+(N-1)*INCX) The vector z. INCX INCX is INTEGER The increment between elements of X, Y and Z. INCX > 0. C C is REAL array, dimension (1+(N-1)*INCC) The cosines of the plane rotations. S S is REAL array, dimension (1+(N-1)*INCC) The sines of the plane rotations. INCC INCC is INTEGER The increment between elements of C and S. INCC > 0. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slarf (character SIDE, integer M, integer N, real, dimension( * ) V, integer INCV, real TAU, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK) SLARF applies an elementary reflector to a general rectangular matrix. Purpose: SLARF applies a real elementary reflector H to a real m by n matrix C, from either the left or the right. H is represented in the form H = I - tau * v * v**T where tau is a real scalar and v is a real vector. If tau = 0, then H is taken to be the unit matrix. Parameters: SIDE SIDE is CHARACTER*1 = 'L': form H * C = 'R': form C * H M M is INTEGER The number of rows of the matrix C. N N is INTEGER The number of columns of the matrix C. V V is REAL array, dimension (1 + (M-1)*abs(INCV)) if SIDE = 'L' or (1 + (N-1)*abs(INCV)) if SIDE = 'R' The vector v in the representation of H. V is not used if TAU = 0. INCV INCV is INTEGER The increment between elements of v. INCV <> 0. TAU TAU is REAL The value tau in the representation of H. C C is REAL array, dimension (LDC,N) On entry, the m by n matrix C. On exit, C is overwritten by the matrix H * C if SIDE = 'L', or C * H if SIDE = 'R'. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is REAL array, dimension (N) if SIDE = 'L' or (M) if SIDE = 'R' Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slarfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T, integer LDT, real, dimension( ldc, * ) C, integer LDC, real, dimension( ldwork, * ) WORK, integer LDWORK) SLARFB applies a block reflector or its transpose to a general rectangular matrix. Purpose: SLARFB applies a real block reflector H or its transpose H**T to a real m by n matrix C, from either the left or the right. Parameters: SIDE SIDE is CHARACTER*1 = 'L': apply H or H**T from the Left = 'R': apply H or H**T from the Right TRANS TRANS is CHARACTER*1 = 'N': apply H (No transpose) = 'T': apply H**T (Transpose) DIRECT DIRECT is CHARACTER*1 Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': H = H(k) . . . H(2) H(1) (Backward) STOREV STOREV is CHARACTER*1 Indicates how the vectors which define the elementary reflectors are stored: = 'C': Columnwise = 'R': Rowwise M M is INTEGER The number of rows of the matrix C. N N is INTEGER The number of columns of the matrix C. K K is INTEGER The order of the matrix T (= the number of elementary reflectors whose product defines the block reflector). V V is REAL array, dimension (LDV,K) if STOREV = 'C' (LDV,M) if STOREV = 'R' and SIDE = 'L' (LDV,N) if STOREV = 'R' and SIDE = 'R' The matrix V. See Further Details. LDV LDV is INTEGER The leading dimension of the array V. If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M); if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N); if STOREV = 'R', LDV >= K. T T is REAL array, dimension (LDT,K) The triangular k by k matrix T in the representation of the block reflector. LDT LDT is INTEGER The leading dimension of the array T. LDT >= K. C C is REAL array, dimension (LDC,N) On entry, the m by n matrix C. On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T. LDC LDC is INTEGER The leading dimension of the array C. LDC >= max(1,M). WORK WORK is REAL array, dimension (LDWORK,K) LDWORK LDWORK is INTEGER The leading dimension of the array WORK. If SIDE = 'L', LDWORK >= max(1,N); if SIDE = 'R', LDWORK >= max(1,M). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2013 Further Details: The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array elements are modified but restored on exit. The rest of the array is not used. DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) ( v1 1 ) ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) ( v1 v2 v3 ) DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V = ( v1 v2 v3 ) V = ( v1 v1 1 ) ( v1 v2 v3 ) ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) ( 1 ) subroutine slarfg (integer N, real ALPHA, real, dimension( * ) X, integer INCX, real TAU) SLARFG generates an elementary reflector (Householder matrix). Purpose: SLARFG generates a real elementary reflector H of order n, such that H * ( alpha ) = ( beta ), H**T * H = I. ( x ) ( 0 ) where alpha and beta are scalars, and x is an (n-1)-element real vector. H is represented in the form H = I - tau * ( 1 ) * ( 1 v**T ) , ( v ) where tau is a real scalar and v is a real (n-1)-element vector. If the elements of x are all zero, then tau = 0 and H is taken to be the unit matrix. Otherwise 1 <= tau <= 2. Parameters: N N is INTEGER The order of the elementary reflector. ALPHA ALPHA is REAL On entry, the value alpha. On exit, it is overwritten with the value beta. X X is REAL array, dimension (1+(N-2)*abs(INCX)) On entry, the vector x. On exit, it is overwritten with the vector v. INCX INCX is INTEGER The increment between elements of X. INCX > 0. TAU TAU is REAL The value tau. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slarfgp (integer N, real ALPHA, real, dimension( * ) X, integer INCX, real TAU) SLARFGP generates an elementary reflector (Householder matrix) with non-negative beta. Purpose: SLARFGP generates a real elementary reflector H of order n, such that H * ( alpha ) = ( beta ), H**T * H = I. ( x ) ( 0 ) where alpha and beta are scalars, beta is non-negative, and x is an (n-1)-element real vector. H is represented in the form H = I - tau * ( 1 ) * ( 1 v**T ) , ( v ) where tau is a real scalar and v is a real (n-1)-element vector. If the elements of x are all zero, then tau = 0 and H is taken to be the unit matrix. Parameters: N N is INTEGER The order of the elementary reflector. ALPHA ALPHA is REAL On entry, the value alpha. On exit, it is overwritten with the value beta. X X is REAL array, dimension (1+(N-2)*abs(INCX)) On entry, the vector x. On exit, it is overwritten with the vector v. INCX INCX is INTEGER The increment between elements of X. INCX > 0. TAU TAU is REAL The value tau. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slarft (character DIRECT, character STOREV, integer N, integer K, real, dimension( ldv, * ) V, integer LDV, real, dimension( * ) TAU, real, dimension( ldt, * ) T, integer LDT) SLARFT forms the triangular factor T of a block reflector H = I - vtvH Purpose: SLARFT forms the triangular factor T of a real block reflector H of order n, which is defined as a product of k elementary reflectors. If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. If STOREV = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the array V, and H = I - V * T * V**T If STOREV = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array V, and H = I - V**T * T * V Parameters: DIRECT DIRECT is CHARACTER*1 Specifies the order in which the elementary reflectors are multiplied to form the block reflector: = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': H = H(k) . . . H(2) H(1) (Backward) STOREV STOREV is CHARACTER*1 Specifies how the vectors which define the elementary reflectors are stored (see also Further Details): = 'C': columnwise = 'R': rowwise N N is INTEGER The order of the block reflector H. N >= 0. K K is INTEGER The order of the triangular factor T (= the number of elementary reflectors). K >= 1. V V is REAL array, dimension (LDV,K) if STOREV = 'C' (LDV,N) if STOREV = 'R' The matrix V. See further details. LDV LDV is INTEGER The leading dimension of the array V. If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K. TAU TAU is REAL array, dimension (K) TAU(i) must contain the scalar factor of the elementary reflector H(i). T T is REAL array, dimension (LDT,K) The k by k triangular factor T of the block reflector. If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is lower triangular. The rest of the array is not used. LDT LDT is INTEGER The leading dimension of the array T. LDT >= K. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Further Details: The shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the following example with n = 5 and k = 3. The elements equal to 1 are not stored. DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) ( v1 1 ) ( 1 v2 v2 v2 ) ( v1 v2 1 ) ( 1 v3 v3 ) ( v1 v2 v3 ) ( v1 v2 v3 ) DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': V = ( v1 v2 v3 ) V = ( v1 v1 1 ) ( v1 v2 v3 ) ( v2 v2 v2 1 ) ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) ( 1 v3 ) ( 1 ) subroutine slarfx (character SIDE, integer M, integer N, real, dimension( * ) V, real TAU, real, dimension( ldc, * ) C, integer LDC, real, dimension( * ) WORK) SLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10. Purpose: SLARFX applies a real elementary reflector H to a real m by n matrix C, from either the left or the right. H is represented in the form H = I - tau * v * v**T where tau is a real scalar and v is a real vector. If tau = 0, then H is taken to be the unit matrix This version uses inline code if H has order < 11. Parameters: SIDE SIDE is CHARACTER*1 = 'L': form H * C = 'R': form C * H M M is INTEGER The number of rows of the matrix C. N N is INTEGER The number of columns of the matrix C. V V is REAL array, dimension (M) if SIDE = 'L' or (N) if SIDE = 'R' The vector v in the representation of H. TAU TAU is REAL The value tau in the representation of H. C C is REAL array, dimension (LDC,N) On entry, the m by n matrix C. On exit, C is overwritten by the matrix H * C if SIDE = 'L', or C * H if SIDE = 'R'. LDC LDC is INTEGER The leading dimension of the array C. LDA >= (1,M). WORK WORK is REAL array, dimension (N) if SIDE = 'L' or (M) if SIDE = 'R' WORK is not referenced if H has order < 11. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slargv (integer N, real, dimension( * ) X, integer INCX, real, dimension( * ) Y, integer INCY, real, dimension( * ) C, integer INCC) SLARGV generates a vector of plane rotations with real cosines and real sines. Purpose: SLARGV generates a vector of real plane rotations, determined by elements of the real vectors x and y. For i = 1,2,...,n ( c(i) s(i) ) ( x(i) ) = ( a(i) ) ( -s(i) c(i) ) ( y(i) ) = ( 0 ) Parameters: N N is INTEGER The number of plane rotations to be generated. X X is REAL array, dimension (1+(N-1)*INCX) On entry, the vector x. On exit, x(i) is overwritten by a(i), for i = 1,...,n. INCX INCX is INTEGER The increment between elements of X. INCX > 0. Y Y is REAL array, dimension (1+(N-1)*INCY) On entry, the vector y. On exit, the sines of the plane rotations. INCY INCY is INTEGER The increment between elements of Y. INCY > 0. C C is REAL array, dimension (1+(N-1)*INCC) The cosines of the plane rotations. INCC INCC is INTEGER The increment between elements of C. INCC > 0. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slarrv (integer N, real VL, real VU, real, dimension( * ) D, real, dimension( * ) L, real PIVMIN, integer, dimension( * ) ISPLIT, integer M, integer DOL, integer DOU, real MINRGP, real RTOL1, real RTOL2, real, dimension( * ) W, real, dimension( * ) WERR, real, dimension( * ) WGAP, integer, dimension( * ) IBLOCK, integer, dimension( * ) INDEXW, real, dimension( * ) GERS, real, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO) SLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT. Purpose: SLARRV computes the eigenvectors of the tridiagonal matrix T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T. The input eigenvalues should have been computed by SLARRE. Parameters: N N is INTEGER The order of the matrix. N >= 0. VL VL is REAL Lower bound of the interval that contains the desired eigenvalues. VL < VU. Needed to compute gaps on the left or right end of the extremal eigenvalues in the desired RANGE. VU VU is REAL Upper bound of the interval that contains the desired eigenvalues. VL < VU. Note: VU is currently not used by this implementation of SLARRV, VU is passed to SLARRV because it could be used compute gaps on the right end of the extremal eigenvalues. However, with not much initial accuracy in LAMBDA and VU, the formula can lead to an overestimation of the right gap and thus to inadequately early RQI 'convergence'. This is currently prevented this by forcing a small right gap. And so it turns out that VU is currently not used by this implementation of SLARRV. D D is REAL array, dimension (N) On entry, the N diagonal elements of the diagonal matrix D. On exit, D may be overwritten. L L is REAL array, dimension (N) On entry, the (N-1) subdiagonal elements of the unit bidiagonal matrix L are in elements 1 to N-1 of L (if the matrix is not split.) At the end of each block is stored the corresponding shift as given by SLARRE. On exit, L is overwritten. PIVMIN PIVMIN is REAL The minimum pivot allowed in the Sturm sequence. ISPLIT ISPLIT is INTEGER array, dimension (N) The splitting points, at which T breaks up into blocks. The first block consists of rows/columns 1 to ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc. M M is INTEGER The total number of input eigenvalues. 0 <= M <= N. DOL DOL is INTEGER DOU DOU is INTEGER If the user wants to compute only selected eigenvectors from all the eigenvalues supplied, he can specify an index range DOL:DOU. Or else the setting DOL=1, DOU=M should be applied. Note that DOL and DOU refer to the order in which the eigenvalues are stored in W. If the user wants to compute only selected eigenpairs, then the columns DOL-1 to DOU+1 of the eigenvector space Z contain the computed eigenvectors. All other columns of Z are set to zero. MINRGP MINRGP is REAL RTOL1 RTOL1 is REAL RTOL2 RTOL2 is REAL Parameters for bisection. An interval [LEFT,RIGHT] has converged if RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) ) W W is REAL array, dimension (N) The first M elements of W contain the APPROXIMATE eigenvalues for which eigenvectors are to be computed. The eigenvalues should be grouped by split-off block and ordered from smallest to largest within the block ( The output array W from SLARRE is expected here ). Furthermore, they are with respect to the shift of the corresponding root representation for their block. On exit, W holds the eigenvalues of the UNshifted matrix. WERR WERR is REAL array, dimension (N) The first M elements contain the semiwidth of the uncertainty interval of the corresponding eigenvalue in W WGAP WGAP is REAL array, dimension (N) The separation from the right neighbor eigenvalue in W. IBLOCK IBLOCK is INTEGER array, dimension (N) The indices of the blocks (submatrices) associated with the corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to the first block from the top, =2 if W(i) belongs to the second block, etc. INDEXW INDEXW is INTEGER array, dimension (N) The indices of the eigenvalues within each block (submatrix); for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th eigenvalue W(i) is the 10-th eigenvalue in the second block. GERS GERS is REAL array, dimension (2*N) The N Gerschgorin intervals (the i-th Gerschgorin interval is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should be computed from the original UNshifted matrix. Z Z is REAL array, dimension (LDZ, max(1,M) ) If INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix T corresponding to the input eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). Note: the user must ensure that at least max(1,M) columns are supplied in the array Z. LDZ LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). ISUPPZ ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The I-th eigenvector is nonzero only in elements ISUPPZ( 2*I-1 ) through ISUPPZ( 2*I ). WORK WORK is REAL array, dimension (12*N) IWORK IWORK is INTEGER array, dimension (7*N) INFO INFO is INTEGER = 0: successful exit > 0: A problem occurred in SLARRV. < 0: One of the called subroutines signaled an internal problem. Needs inspection of the corresponding parameter IINFO for further information. =-1: Problem in SLARRB when refining a child's eigenvalues. =-2: Problem in SLARRF when computing the RRR of a child. When a child is inside a tight cluster, it can be difficult to find an RRR. A partial remedy from the user's point of view is to make the parameter MINRGP smaller and recompile. However, as the orthogonality of the computed vectors is proportional to 1/MINRGP, the user should be aware that he might be trading in precision when he decreases MINRGP. =-3: Problem in SLARRB when refining a single eigenvalue after the Rayleigh correction was rejected. = 5: The Rayleigh Quotient Iteration failed to converge to full accuracy in MAXITR steps. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2016 Contributors: Beresford Parlett, University of California, Berkeley, USA Jim Demmel, University of California, Berkeley, USA Inderjit Dhillon, University of Texas, Austin, USA Osni Marques, LBNL/NERSC, USA Christof Voemel, University of California, Berkeley, USA subroutine slartv (integer N, real, dimension( * ) X, integer INCX, real, dimension( * ) Y, integer INCY, real, dimension( * ) C, real, dimension( * ) S, integer INCC) SLARTV applies a vector of plane rotations with real cosines and real sines to the elements of a pair of vectors. Purpose: SLARTV applies a vector of real plane rotations to elements of the real vectors x and y. For i = 1,2,...,n ( x(i) ) := ( c(i) s(i) ) ( x(i) ) ( y(i) ) ( -s(i) c(i) ) ( y(i) ) Parameters: N N is INTEGER The number of plane rotations to be applied. X X is REAL array, dimension (1+(N-1)*INCX) The vector x. INCX INCX is INTEGER The increment between elements of X. INCX > 0. Y Y is REAL array, dimension (1+(N-1)*INCY) The vector y. INCY INCY is INTEGER The increment between elements of Y. INCY > 0. C C is REAL array, dimension (1+(N-1)*INCC) The cosines of the plane rotations. S S is REAL array, dimension (1+(N-1)*INCC) The sines of the plane rotations. INCC INCC is INTEGER The increment between elements of C and S. INCC > 0. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slaswp (integer N, real, dimension( lda, * ) A, integer LDA, integer K1, integer K2, integer, dimension( * ) IPIV, integer INCX) SLASWP performs a series of row interchanges on a general rectangular matrix. Purpose: SLASWP performs a series of row interchanges on the matrix A. One row interchange is initiated for each of rows K1 through K2 of A. Parameters: N N is INTEGER The number of columns of the matrix A. A A is REAL array, dimension (LDA,N) On entry, the matrix of column dimension N to which the row interchanges will be applied. On exit, the permuted matrix. LDA LDA is INTEGER The leading dimension of the array A. K1 K1 is INTEGER The first element of IPIV for which a row interchange will be done. K2 K2 is INTEGER (K2-K1+1) is the number of elements of IPIV for which a row interchange will be done. IPIV IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX)) The vector of pivot indices. Only the elements in positions K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed. IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be interchanged. INCX INCX is INTEGER The increment between successive values of IPIV. If INCX is negative, the pivots are applied in reverse order. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2017 Further Details: Modified by R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA subroutine slatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, integer KD, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO) SLATBS solves a triangular banded system of equations. Purpose: SLATBS solves one of the triangular systems A *x = s*b or A**T*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular band matrix. Here A**T denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine STBSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. Parameters: UPLO UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular TRANS TRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T* x = s*b (Transpose) = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) DIAG DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular NORMIN NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM. N N is INTEGER The order of the matrix A. N >= 0. KD KD is INTEGER The number of subdiagonals or superdiagonals in the triangular matrix A. KD >= 0. AB AB is REAL array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). LDAB LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1. X X is REAL array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x. SCALE SCALE is REAL The scaling factor s for the triangular system A * x = s*b or A**T* x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0. CNORM CNORM is REAL array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Further Details: A rough bound on x is computed; if that is less than overflow, STBSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine STBSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow). The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A**T*x = b. The basic algorithm for A upper triangular is for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call STBSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow). subroutine slatdf (integer IJOB, integer N, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) RHS, real RDSUM, real RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV) SLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate. Purpose: SLATDF uses the LU factorization of the n-by-n matrix Z computed by SGETC2 and computes a contribution to the reciprocal Dif-estimate by solving Z * x = b for x, and choosing the r.h.s. b such that the norm of x is as large as possible. On entry RHS = b holds the contribution from earlier solved sub-systems, and on return RHS = x. The factorization of Z returned by SGETC2 has the form Z = P*L*U*Q, where P and Q are permutation matrices. L is lower triangular with unit diagonal elements and U is upper triangular. Parameters: IJOB IJOB is INTEGER IJOB = 2: First compute an approximative null-vector e of Z using SGECON, e is normalized and solve for Zx = +-e - f with the sign giving the greater value of 2-norm(x). About 5 times as expensive as Default. IJOB .ne. 2: Local look ahead strategy where all entries of the r.h.s. b is chosen as either +1 or -1 (Default). N N is INTEGER The number of columns of the matrix Z. Z Z is REAL array, dimension (LDZ, N) On entry, the LU part of the factorization of the n-by-n matrix Z computed by SGETC2: Z = P * L * U * Q LDZ LDZ is INTEGER The leading dimension of the array Z. LDA >= max(1, N). RHS RHS is REAL array, dimension N. On entry, RHS contains contributions from other subsystems. On exit, RHS contains the solution of the subsystem with entries acoording to the value of IJOB (see above). RDSUM RDSUM is REAL On entry, the sum of squares of computed contributions to the Dif-estimate under computation by STGSYL, where the scaling factor RDSCAL (see below) has been factored out. On exit, the corresponding sum of squares updated with the contributions from the current sub-system. If TRANS = 'T' RDSUM is not touched. NOTE: RDSUM only makes sense when STGSY2 is called by STGSYL. RDSCAL RDSCAL is REAL On entry, scaling factor used to prevent overflow in RDSUM. On exit, RDSCAL is updated w.r.t. the current contributions in RDSUM. If TRANS = 'T', RDSCAL is not touched. NOTE: RDSCAL only makes sense when STGSY2 is called by STGSYL. IPIV IPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= i <= N, row i of the matrix has been interchanged with row IPIV(i). JPIV JPIV is INTEGER array, dimension (N). The pivot indices; for 1 <= j <= N, column j of the matrix has been interchanged with column JPIV(j). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: June 2016 Further Details: This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization. Contributors: Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. References: [1] Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751. [2] Peter Poromaa, On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. Report IMINF-95.05, Departement of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995. subroutine slatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, real, dimension( * ) AP, real, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO) SLATPS solves a triangular system of equations with the matrix held in packed storage. Purpose: SLATPS solves one of the triangular systems A *x = s*b or A**T*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form. Here A**T denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine STPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. Parameters: UPLO UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular TRANS TRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T* x = s*b (Transpose) = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) DIAG DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular NORMIN NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM. N N is INTEGER The order of the matrix A. N >= 0. AP AP is REAL array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. X X is REAL array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x. SCALE SCALE is REAL The scaling factor s for the triangular system A * x = s*b or A**T* x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0. CNORM CNORM is REAL array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Further Details: A rough bound on x is computed; if that is less than overflow, STPSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine STPSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow). The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A**T*x = b. The basic algorithm for A upper triangular is for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call STPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow). subroutine slatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO) SLATRS solves a triangular system of equations with the scale factor set to prevent overflow. Purpose: SLATRS solves one of the triangular systems A *x = s*b or A**T*x = s*b with scaling to prevent overflow. Here A is an upper or lower triangular matrix, A**T denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine STRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned. Parameters: UPLO UPLO is CHARACTER*1 Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular = 'L': Lower triangular TRANS TRANS is CHARACTER*1 Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose) = 'T': Solve A**T* x = s*b (Transpose) = 'C': Solve A**T* x = s*b (Conjugate transpose = Transpose) DIAG DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular = 'U': Unit triangular NORMIN NORMIN is CHARACTER*1 Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM. N N is INTEGER The order of the matrix A. N >= 0. A A is REAL array, dimension (LDA,N) The triangular matrix A. If UPLO = 'U', the leading n by n upper triangular part of the array A contains the upper triangular matrix, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n by n lower triangular part of the array A contains the lower triangular matrix, and the strictly upper triangular part of A is not referenced. If DIAG = 'U', the diagonal elements of A are also not referenced and are assumed to be 1. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max (1,N). X X is REAL array, dimension (N) On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x. SCALE SCALE is REAL The scaling factor s for the triangular system A * x = s*b or A**T* x = s*b. If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0. CNORM CNORM is REAL array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A. INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Further Details: A rough bound on x is computed; if that is less than overflow, STRSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation. A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is x[1:n] := b[1:n] for j = 1, ..., n x(j) := x(j) / A(j,j) x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j] end Define bounds on the components of x after j iterations of the loop: M(j) = bound on x[1:j] G(j) = bound on x[j+1:n] Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}. Then for iteration j+1 we have M(j+1) <= G(j) / | A(j+1,j+1) | G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] | <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | ) where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | ) 1<=i<=j and |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| ) 1<=i< j Since |x(j)| <= M(j), we use the Level 2 BLAS routine STRSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow). The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve A**T*x = b. The basic algorithm for A upper triangular is for j = 1, ..., n x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i) - A[1:i-1,i]**T * x[1:i-1] ), 1<=i<=j M(j) = bound on x(i), 1<=i<=j The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) | <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| ) 1<=i<=j and we can safely call STRSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow). subroutine slauu2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO) SLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm). Purpose: SLAUU2 computes the product U * U**T or L**T * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in A. If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in A. This is the unblocked form of the algorithm, calling Level 2 BLAS. Parameters: UPLO UPLO is CHARACTER*1 Specifies whether the triangular factor stored in the array A is upper or lower triangular: = 'U': Upper triangular = 'L': Lower triangular N N is INTEGER The order of the triangular factor U or L. N >= 0. A A is REAL array, dimension (LDA,N) On entry, the triangular factor U or L. On exit, if UPLO = 'U', the upper triangle of A is overwritten with the upper triangle of the product U * U**T; if UPLO = 'L', the lower triangle of A is overwritten with the lower triangle of the product L**T * L. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine slauum (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO) SLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm). Purpose: SLAUUM computes the product U * U**T or L**T * L, where the triangular factor U or L is stored in the upper or lower triangular part of the array A. If UPLO = 'U' or 'u' then the upper triangle of the result is stored, overwriting the factor U in A. If UPLO = 'L' or 'l' then the lower triangle of the result is stored, overwriting the factor L in A. This is the blocked form of the algorithm, calling Level 3 BLAS. Parameters: UPLO UPLO is CHARACTER*1 Specifies whether the triangular factor stored in the array A is upper or lower triangular: = 'U': Upper triangular = 'L': Lower triangular N N is INTEGER The order of the triangular factor U or L. N >= 0. A A is REAL array, dimension (LDA,N) On entry, the triangular factor U or L. On exit, if UPLO = 'U', the upper triangle of A is overwritten with the upper triangle of the product U * U**T; if UPLO = 'L', the lower triangle of A is overwritten with the lower triangle of the product L**T * L. LDA LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N). INFO INFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal value Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine srscl (integer N, real SA, real, dimension( * ) SX, integer INCX) SRSCL multiplies a vector by the reciprocal of a real scalar. Purpose: SRSCL multiplies an n-element real vector x by the real scalar 1/a. This is done without overflow or underflow as long as the final result x/a does not overflow or underflow. Parameters: N N is INTEGER The number of components of the vector x. SA SA is REAL The scalar a which is used to divide each component of x. SA must be >= 0, or the subroutine will divide by zero. SX SX is REAL array, dimension (1+(N-1)*abs(INCX)) The n-element vector x. INCX INCX is INTEGER The increment between successive values of the vector SX. > 0: SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i), 1< i<= n Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 subroutine stprfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldt, * ) T, integer LDT, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldwork, * ) WORK, integer LDWORK) STPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks. Purpose: STPRFB applies a real "triangular-pentagonal" block reflector H or its conjugate transpose H^H to a real matrix C, which is composed of two blocks A and B, either from the left or right. Parameters: SIDE SIDE is CHARACTER*1 = 'L': apply H or H^H from the Left = 'R': apply H or H^H from the Right TRANS TRANS is CHARACTER*1 = 'N': apply H (No transpose) = 'C': apply H^H (Conjugate transpose) DIRECT DIRECT is CHARACTER*1 Indicates how H is formed from a product of elementary reflectors = 'F': H = H(1) H(2) . . . H(k) (Forward) = 'B': H = H(k) . . . H(2) H(1) (Backward) STOREV STOREV is CHARACTER*1 Indicates how the vectors which define the elementary reflectors are stored: = 'C': Columns = 'R': Rows M M is INTEGER The number of rows of the matrix B. M >= 0. N N is INTEGER The number of columns of the matrix B. N >= 0. K K is INTEGER The order of the matrix T, i.e. the number of elementary reflectors whose product defines the block reflector. K >= 0. L L is INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details. V V is REAL array, dimension (LDV,K) if STOREV = 'C' (LDV,M) if STOREV = 'R' and SIDE = 'L' (LDV,N) if STOREV = 'R' and SIDE = 'R' The pentagonal matrix V, which contains the elementary reflectors H(1), H(2), ..., H(K). See Further Details. LDV LDV is INTEGER The leading dimension of the array V. If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M); if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N); if STOREV = 'R', LDV >= K. T T is REAL array, dimension (LDT,K) The triangular K-by-K matrix T in the representation of the block reflector. LDT LDT is INTEGER The leading dimension of the array T. LDT >= K. A A is REAL array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of H*C or H^H*C or C*H or C*H^H. See Further Details. LDA LDA is INTEGER The leading dimension of the array A. If SIDE = 'L', LDC >= max(1,K); If SIDE = 'R', LDC >= max(1,M). B B is REAL array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of H*C or H^H*C or C*H or C*H^H. See Further Details. LDB LDB is INTEGER The leading dimension of the array B. LDB >= max(1,M). WORK WORK is REAL array, dimension (LDWORK,N) if SIDE = 'L', (LDWORK,K) if SIDE = 'R'. LDWORK LDWORK is INTEGER The leading dimension of the array WORK. If SIDE = 'L', LDWORK >= K; if SIDE = 'R', LDWORK >= M. Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: December 2016 Further Details: The matrix C is a composite matrix formed from blocks A and B. The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K, and if SIDE = 'L', A is of size K-by-N. If SIDE = 'R' and DIRECT = 'F', C = [A B]. If SIDE = 'L' and DIRECT = 'F', C = [A] [B]. If SIDE = 'R' and DIRECT = 'B', C = [B A]. If SIDE = 'L' and DIRECT = 'B', C = [B] [A]. The pentagonal matrix V is composed of a rectangular block V1 and a trapezoidal block V2. The size of the trapezoidal block is determined by the parameter L, where 0<=L<=K. If L=K, the V2 block of V is triangular; if L=0, there is no trapezoidal block, thus V = V1 is rectangular. If DIRECT = 'F' and STOREV = 'C': V = [V1] [V2] - V2 is upper trapezoidal (first L rows of K-by-K upper triangular) If DIRECT = 'F' and STOREV = 'R': V = [V1 V2] - V2 is lower trapezoidal (first L columns of K-by-K lower triangular) If DIRECT = 'B' and STOREV = 'C': V = [V2] [V1] - V2 is lower trapezoidal (last L rows of K-by-K lower triangular) If DIRECT = 'B' and STOREV = 'R': V = [V2 V1] - V2 is upper trapezoidal (last L columns of K-by-K upper triangular) If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K. If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K. If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L. If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L. 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Version 3.7.1 Tue Nov 28 2017 realOTHERauxiliary(3)
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