CGEBD2(3F) CGEBD2(3F)
CGEBD2 - reduce a complex general m by n matrix A to upper or lower real
bidiagonal form B by a unitary transformation
SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
INTEGER INFO, LDA, M, N
REAL D( * ), E( * )
COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
CGEBD2 reduces a complex general m by n matrix A to upper or lower real
bidiagonal form B by a unitary transformation: Q' * 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) COMPLEX 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 unitary matrix Q as
a product of elementary reflectors, and the elements above the
first superdiagonal, with the array TAUP, represent the unitary
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 unitary matrix Q as a product
of elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the unitary 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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CGEBD2(3F) CGEBD2(3F)
TAUQ (output) COMPLEX array dimension (min(M,N))
The scalar factors of the elementary reflectors which represent
the unitary matrix Q. See Further Details. TAUP (output)
COMPLEX array, dimension (min(M,N)) The scalar factors of the
elementary reflectors which represent the unitary matrix P. See
Further Details. WORK (workspace) COMPLEX array, dimension
(max(M,N))
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 complex scalars, and v and u are complex 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 complex scalars, v and u are complex 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 )
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CGEBD2(3F) CGEBD2(3F)
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).
CGEBD2(3F) CGEBD2(3F)
CGEBD2 - reduce a complex general m by n matrix A to upper or lower real
bidiagonal form B by a unitary transformation
SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
INTEGER INFO, LDA, M, N
REAL D( * ), E( * )
COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
CGEBD2 reduces a complex general m by n matrix A to upper or lower real
bidiagonal form B by a unitary transformation: Q' * 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) COMPLEX 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 unitary matrix Q as
a product of elementary reflectors, and the elements above the
first superdiagonal, with the array TAUP, represent the unitary
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 unitary matrix Q as a product
of elementary reflectors, and the elements above the diagonal,
with the array TAUP, represent the unitary 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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CGEBD2(3F) CGEBD2(3F)
TAUQ (output) COMPLEX array dimension (min(M,N))
The scalar factors of the elementary reflectors which represent
the unitary matrix Q. See Further Details. TAUP (output)
COMPLEX array, dimension (min(M,N)) The scalar factors of the
elementary reflectors which represent the unitary matrix P. See
Further Details. WORK (workspace) COMPLEX array, dimension
(max(M,N))
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 complex scalars, and v and u are complex 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 complex scalars, v and u are complex 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 )
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CGEBD2(3F) CGEBD2(3F)
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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