ZLABRD(3F) ZLABRD(3F)
ZLABRD - reduce 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' * A * P, and returns the matrices X and Y which are
needed to apply the transformation to the unreduced part of A
SUBROUTINE ZLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )
INTEGER LDA, LDX, LDY, M, N, NB
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y(
LDY, * )
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' * 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 (input) INTEGER
The number of rows in the matrix A.
N (input) INTEGER
The number of columns in the matrix A.
NB (input) INTEGER
The number of leading rows and columns of A to be reduced.
A (input/output) 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 (input) INTEGER The leading dimension
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ZLABRD(3F) ZLABRD(3F)
of the array A. LDA >= max(1,M).
D (output) 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 (output) DOUBLE PRECISION array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of the
reduced matrix.
TAUQ (output) COMPLEX*16 array dimension (NB)
The scalar factors of the elementary reflectors which represent
the unitary matrix Q. See Further Details. TAUP (output)
COMPLEX*16 array, dimension (NB) The scalar factors of the
elementary reflectors which represent the unitary matrix P. See
Further Details. X (output) COMPLEX*16 array, dimension
(LDX,NB) The m-by-nb matrix X required to update the unreduced
part of A.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,M).
Y (output) COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part of A.
LDY (output) INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
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' and G(i) = I - taup * u * u'
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' 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' - X*U'.
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ZLABRD(3F) ZLABRD(3F)
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).
ZLABRD(3F) ZLABRD(3F)
ZLABRD - reduce 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' * A * P, and returns the matrices X and Y which are
needed to apply the transformation to the unreduced part of A
SUBROUTINE ZLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY )
INTEGER LDA, LDX, LDY, M, N, NB
DOUBLE PRECISION D( * ), E( * )
COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), X( LDX, * ), Y(
LDY, * )
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' * 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 (input) INTEGER
The number of rows in the matrix A.
N (input) INTEGER
The number of columns in the matrix A.
NB (input) INTEGER
The number of leading rows and columns of A to be reduced.
A (input/output) 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 (input) INTEGER The leading dimension
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ZLABRD(3F) ZLABRD(3F)
of the array A. LDA >= max(1,M).
D (output) 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 (output) DOUBLE PRECISION array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of the
reduced matrix.
TAUQ (output) COMPLEX*16 array dimension (NB)
The scalar factors of the elementary reflectors which represent
the unitary matrix Q. See Further Details. TAUP (output)
COMPLEX*16 array, dimension (NB) The scalar factors of the
elementary reflectors which represent the unitary matrix P. See
Further Details. X (output) COMPLEX*16 array, dimension
(LDX,NB) The m-by-nb matrix X required to update the unreduced
part of A.
LDX (input) INTEGER
The leading dimension of the array X. LDX >= max(1,M).
Y (output) COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part of A.
LDY (output) INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
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' and G(i) = I - taup * u * u'
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' 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' - X*U'.
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ZLABRD(3F) ZLABRD(3F)
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).
PPPPaaaaggggeeee 3333 [ Back ]
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