SGEBRD(3F) SGEBRD(3F)
SGEBRD - reduce a general real M-by-N matrix A to upper or lower
bidiagonal form B by an orthogonal transformation
SUBROUTINE SGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )
INTEGER INFO, LDA, LWORK, M, N
REAL A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ),
WORK( LWORK )
SGEBRD reduces a general real M-by-N matrix A to upper or lower
bidiagonal 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.
M (input) INTEGER
The number of rows in the matrix A. M >= 0.
N (input) INTEGER
The number of columns in the matrix A. N >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N general matrix to be reduced. On exit, if m
>= n, the diagonal and the first superdiagonal are overwritten
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 overwritten 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 reflectors, and the elements above
the diagonal, with the array TAUP, represent the orthogonal
matrix P as a product of elementary reflectors. See Further
Details. LDA (input) INTEGER The leading dimension of the
array A. LDA >= max(1,M).
D (output) REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
E (output) REAL array, dimension (min(M,N)-1)
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.
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SGEBRD(3F) SGEBRD(3F)
TAUQ (output) REAL array dimension (min(M,N))
The scalar factors of the elementary reflectors which represent
the orthogonal matrix Q. See Further Details. TAUP (output)
REAL array, dimension (min(M,N)) The scalar factors of the
elementary reflectors which represent the orthogonal matrix P.
See Further Details. WORK (workspace/output) REAL array,
dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the
optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,M,N). For optimum
performance LWORK >= (M+N)*NB, where NB is the optimal blocksize.
INFO (output) INTEGER
= 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
reflectors:
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 )
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SGEBRD(3F) SGEBRD(3F)
( 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).
SGEBRD(3F) SGEBRD(3F)
SGEBRD - reduce a general real M-by-N matrix A to upper or lower
bidiagonal form B by an orthogonal transformation
SUBROUTINE SGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, INFO )
INTEGER INFO, LDA, LWORK, M, N
REAL A( LDA, * ), D( * ), E( * ), TAUP( * ), TAUQ( * ),
WORK( LWORK )
SGEBRD reduces a general real M-by-N matrix A to upper or lower
bidiagonal 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.
M (input) INTEGER
The number of rows in the matrix A. M >= 0.
N (input) INTEGER
The number of columns in the matrix A. N >= 0.
A (input/output) REAL array, dimension (LDA,N)
On entry, the M-by-N general matrix to be reduced. On exit, if m
>= n, the diagonal and the first superdiagonal are overwritten
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 overwritten 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 reflectors, and the elements above
the diagonal, with the array TAUP, represent the orthogonal
matrix P as a product of elementary reflectors. See Further
Details. LDA (input) INTEGER The leading dimension of the
array A. LDA >= max(1,M).
D (output) REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B: D(i) = A(i,i).
E (output) REAL array, dimension (min(M,N)-1)
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.
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SGEBRD(3F) SGEBRD(3F)
TAUQ (output) REAL array dimension (min(M,N))
The scalar factors of the elementary reflectors which represent
the orthogonal matrix Q. See Further Details. TAUP (output)
REAL array, dimension (min(M,N)) The scalar factors of the
elementary reflectors which represent the orthogonal matrix P.
See Further Details. WORK (workspace/output) REAL array,
dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the
optimal LWORK.
LWORK (input) INTEGER
The length of the array WORK. LWORK >= max(1,M,N). For optimum
performance LWORK >= (M+N)*NB, where NB is the optimal blocksize.
INFO (output) INTEGER
= 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
reflectors:
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 )
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SGEBRD(3F) SGEBRD(3F)
( 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).
PPPPaaaaggggeeee 3333 [ Back ]
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