dgebrd - agonal form B by an orthogonal transformation
SUBROUTINE DGEBRD(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) INTEGER M, N, LDA, LWORK, INFO DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUQ(*), TAUP(*), WORK(*) SUBROUTINE DGEBRD_64(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) INTEGER*8 M, N, LDA, LWORK, INFO DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUQ(*), TAUP(*), WORK(*) F95 INTERFACE SUBROUTINE GEBRD(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) INTEGER :: M, N, LDA, LWORK, INFO REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK REAL(8), DIMENSION(:,:) :: A SUBROUTINE GEBRD_64(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) INTEGER(8) :: M, N, LDA, LWORK, INFO REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK REAL(8), DIMENSION(:,:) :: A C INTERFACE #include <sunperf.h> void dgebrd(int m, int n, double *a, int lda, double *d, double *e, double *tauq, double *taup, int *info); void dgebrd_64(long m, long n, double *a, long lda, double *d, double *e, double *tauq, double *taup, long *info);
Oracle Solaris Studio Performance Library dgebrd(3P) NAME dgebrd - reduce a general real M-by-N matrix A to upper or lower bidi- agonal form B by an orthogonal transformation SYNOPSIS SUBROUTINE DGEBRD(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) INTEGER M, N, LDA, LWORK, INFO DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUQ(*), TAUP(*), WORK(*) SUBROUTINE DGEBRD_64(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) INTEGER*8 M, N, LDA, LWORK, INFO DOUBLE PRECISION A(LDA,*), D(*), E(*), TAUQ(*), TAUP(*), WORK(*) F95 INTERFACE SUBROUTINE GEBRD(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) INTEGER :: M, N, LDA, LWORK, INFO REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK REAL(8), DIMENSION(:,:) :: A SUBROUTINE GEBRD_64(M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO) INTEGER(8) :: M, N, LDA, LWORK, INFO REAL(8), DIMENSION(:) :: D, E, TAUQ, TAUP, WORK REAL(8), DIMENSION(:,:) :: A C INTERFACE #include <sunperf.h> void dgebrd(int m, int n, double *a, int lda, double *d, double *e, double *tauq, double *taup, int *info); void dgebrd_64(long m, long n, double *a, long lda, double *d, double *e, double *tauq, double *taup, long *info); PURPOSE dgebrd reduces a general real M-by-N matrix A to upper or lower bidiag- onal form B by an orthogonal transformation: Q**T * A * P = B. If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. ARGUMENTS M (input) The number of rows in the matrix A. M >= 0. N (input) The number of columns in the matrix A. N >= 0. A (input/output) On entry, the M-by-N general matrix to be reduced. On exit, if m >= n, the diagonal and the first superdiagonal are over- written with the upper bidiagonal matrix B; the elements below the diagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflectors, and the elements above the first superdiagonal, with the array TAUP, represent the orthogonal matrix P as a product of elementary reflectors; if m < n, the diagonal and the first subdiagonal are over- written with the lower bidiagonal matrix B; the elements below the first subdiagonal, with the array TAUQ, represent the orthogonal matrix Q as a product of elementary reflec- tors, and the elements above the diagonal, with the array TAUP, represent the orthogonal matrix P as a product of ele- mentary reflectors. See Further Details. LDA (input) The leading dimension of the array A. LDA >= max(1,M). D (output) The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i). E (output) The off-diagonal elements of the bidiagonal matrix B: if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. TAUQ (output) The scalar factors of the elementary reflectors which repre- sent the orthogonal matrix Q. See Further Details. TAUP (output) The scalar factors of the elementary reflectors which repre- sent the orthogonal matrix P. See Further Details. WORK (workspace) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) The length of the array WORK. LWORK >= max(1,M,N). For opti- mum performance LWORK >= (M+N)*NB, where NB is the optimal blocksize. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. INFO (output) = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value. FURTHER DETAILS The matrices Q and P are represented as products of elementary reflec- tors: If m >= n, Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) Each H(i) and G(i) has the form: H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). If m < n, Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) Each H(i) and G(i) has the form: H(i) = I - tauq * v * v' and G(i) = I - taup * u * u' where tauq and taup are real scalars, and v and u are real vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); u(1:i-1) = 0, u(i) = 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). The contents of A on exit are illustrated by the following examples: m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) where d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining H(i), and ui an element of the vector defining G(i). 7 Nov 2015 dgebrd(3P)