
NAMEzlabrd.f SYNOPSISFunctions/Subroutinessubroutine zlabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY) Function/Subroutine Documentationsubroutine zlabrd (integerM, integerN, integerNB, complex*16, dimension( lda, * )A, integerLDA, double precision, dimension( * )D, double precision, dimension( * )E, complex*16, dimension( * )TAUQ, complex*16, dimension( * )TAUP, complex*16, dimension( ldx, * )X, integerLDX, complex*16, dimension( ldy, * )Y, integerLDY)ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form. Purpose:ZLABRD reduces the first NB rows and columns of a complex general m by n matrix A to upper or lower real bidiagonal form by a unitary transformation Q**H * 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 ZGEBRD M
Author:
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 COMPLEX*16 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 unitary matrix Q as a product of elementary reflectors; and elements above the diagonal in the first NB rows, with the array TAUP, represent the unitary 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 unitary 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 unitary 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB) The offdiagonal elements of the first NB rows and columns of the reduced matrix.TAUQ TAUQ is COMPLEX*16 array dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details.TAUP TAUP is COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details.X X is COMPLEX*16 array, dimension (LDX,NB) The mbynb 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 COMPLEX*16 array, dimension (LDY,NB) The nbynb 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). Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012
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**H and G(i) = I  taup * u * u**H where tauq and taup are complex scalars, and v and u are complex vectors. If m >= n, v(1:i1) = 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:i1) = 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 mbynb matrix V and the nbbyn matrix U**H 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**H  X*U**H. 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). AuthorGenerated automatically by Doxygen for LAPACK from the source code.
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