LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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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.
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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
[in] | M | M is INTEGER The number of rows in the matrix A. |
[in] | N | N is INTEGER The number of columns in the matrix A. |
[in] | NB | NB is INTEGER The number of leading rows and columns of A to be reduced. |
[in,out] | 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. |
[in] | LDA | LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). |
[out] | 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). |
[out] | E | E is REAL array, dimension (NB) The off-diagonal elements of the first NB rows and columns of the reduced matrix. |
[out] | TAUQ | TAUQ is REAL array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix Q. See Further Details. |
[out] | TAUP | TAUP is REAL array, dimension (NB) The scalar factors of the elementary reflectors which represent the orthogonal matrix P. See Further Details. |
[out] | X | X is REAL array, dimension (LDX,NB) The m-by-nb matrix X required to update the unreduced part of A. |
[in] | LDX | LDX is INTEGER The leading dimension of the array X. LDX >= max(1,M). |
[out] | Y | Y is REAL array, dimension (LDY,NB) The n-by-nb matrix Y required to update the unreduced part of A. |
[in] | LDY | LDY is INTEGER The leading dimension of the array Y. LDY >= max(1,N). |
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).