.TH "SRC/clabrd.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/clabrd.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBclabrd\fP (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)" .br .RI "\fBCLABRD\fP reduces the first nb rows and columns of a general matrix to a bidiagonal form\&. " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine clabrd (integer m, integer n, integer nb, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension( ldx, * ) x, integer ldx, complex, dimension( ldy, * ) y, integer ldy)" .PP \fBCLABRD\fP reduces the first nb rows and columns of a general matrix to a bidiagonal form\&. .PP \fBPurpose:\fP .RS 4 .PP .nf CLABRD 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 CGEBRD .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The number of rows in the matrix A\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns in the matrix A\&. .fi .PP .br \fINB\fP .PP .nf NB is INTEGER The number of leading rows and columns of A to be reduced\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX 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\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .br \fID\fP .PP .nf 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)\&. .fi .PP .br \fIE\fP .PP .nf E is REAL array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix\&. .fi .PP .br \fITAUQ\fP .PP .nf TAUQ is COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix Q\&. See Further Details\&. .fi .PP .br \fITAUP\fP .PP .nf TAUP is COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors which represent the unitary matrix P\&. See Further Details\&. .fi .PP .br \fIX\fP .PP .nf X is COMPLEX array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A\&. .fi .PP .br \fILDX\fP .PP .nf LDX is INTEGER The leading dimension of the array X\&. LDX >= max(1,M)\&. .fi .PP .br \fIY\fP .PP .nf Y is COMPLEX array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A\&. .fi .PP .br \fILDY\fP .PP .nf LDY is INTEGER The leading dimension of the array Y\&. LDY >= max(1,N)\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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: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**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)\&. .fi .PP .RE .PP .PP Definition at line \fB210\fP of file \fBclabrd\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.