Man page - gebrd(3)

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Manual

gebrd

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cgebrd (integer m, integer n, complex, dimension( lda, * ) a,integer lda, real, dimension( * ) d, real, dimension( * ) e, complex,dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension(* ) work, integer lwork, integer info)
subroutine dgebrd (integer m, integer n, double precision, dimension( lda,* ) a, integer lda, double precision, dimension( * ) d, doubleprecision, dimension( * ) e, double precision, dimension( * ) tauq,double precision, dimension( * ) taup, double precision, dimension( * )work, integer lwork, integer info)
subroutine sgebrd (integer m, integer n, real, dimension( lda, * ) a,integer lda, real, dimension( * ) d, real, dimension( * ) e, real,dimension( * ) tauq, real, dimension( * ) taup, real, dimension( * )work, integer lwork, integer info)
subroutine zgebrd (integer m, integer n, complex*16, dimension( lda, * ) a,integer lda, double precision, dimension( * ) d, double precision,dimension( * ) e, complex*16, dimension( * ) tauq, complex*16,dimension( * ) taup, complex*16, dimension( * ) work, integer lwork,integer info)
Author

NAME

gebrd - gebrd: reduction to bidiagonal

SYNOPSIS

Functions

subroutine cgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
CGEBRD
subroutine dgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
DGEBRD
subroutine sgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
SGEBRD
subroutine zgebrd (m, n, a, lda, d, e, tauq, taup, work, lwork, info)
ZGEBRD

Detailed Description

Function Documentation

subroutine cgebrd (integer m, integer n, complex, dimension( lda, * ) a,integer lda, real, dimension( * ) d, real, dimension( * ) e, complex,dimension( * ) tauq, complex, dimension( * ) taup, complex, dimension(* ) work, integer lwork, integer info)

CGEBRD

Purpose:

CGEBRD reduces a general complex M-by-N matrix A to upper or lower
bidiagonal form B by a unitary transformation: Q**H * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

Parameters

M

M is INTEGER
The number of rows in the matrix A. M >= 0.

N

N is INTEGER
The number of columns in the matrix A. N >= 0.

A

A is 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

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

D

D is REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).

E

E is 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.

TAUQ

TAUQ is COMPLEX array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.

TAUP

TAUP is COMPLEX array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.

WORK

WORK is COMPLEX array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
For optimum performance LWORK >= (M+N)*NB, where NB
is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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**H and G(i) = I - taup * u * u**H

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**H and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and 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 )

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).

subroutine dgebrd (integer m, integer n, double precision, dimension( lda,* ) a, integer lda, double precision, dimension( * ) d, doubleprecision, dimension( * ) e, double precision, dimension( * ) tauq,double precision, dimension( * ) taup, double precision, dimension( * )work, integer lwork, integer info)

DGEBRD

Purpose:

DGEBRD 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.

Parameters

M

M is INTEGER
The number of rows in the matrix A. M >= 0.

N

N is INTEGER
The number of columns in the matrix A. N >= 0.

A

A is DOUBLE PRECISION 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

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

D

D is DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).

E

E is DOUBLE PRECISION 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.

TAUQ

TAUQ is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details.

TAUP

TAUP is DOUBLE PRECISION array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details.

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
For optimum performance LWORK >= (M+N)*NB, where NB
is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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**T and G(i) = I - taup * u * u**T

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**T and G(i) = I - taup * u * u**T

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 )
( 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).

subroutine sgebrd (integer m, integer n, real, dimension( lda, * ) a,integer lda, real, dimension( * ) d, real, dimension( * ) e, real,dimension( * ) tauq, real, dimension( * ) taup, real, dimension( * )work, integer lwork, integer info)

SGEBRD

Purpose:

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.

Parameters

M

M is INTEGER
The number of rows in the matrix A. M >= 0.

N

N is INTEGER
The number of columns in the matrix A. N >= 0.

A

A is 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

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

D

D is REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).

E

E is 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.

TAUQ

TAUQ is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details.

TAUP

TAUP is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details.

WORK

WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
For optimum performance LWORK >= (M+N)*NB, where NB
is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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**T and G(i) = I - taup * u * u**T

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**T and G(i) = I - taup * u * u**T

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 )
( 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).

subroutine zgebrd (integer m, integer n, complex*16, dimension( lda, * ) a,integer lda, double precision, dimension( * ) d, double precision,dimension( * ) e, complex*16, dimension( * ) tauq, complex*16,dimension( * ) taup, complex*16, dimension( * ) work, integer lwork,integer info)

ZGEBRD

Purpose:

ZGEBRD reduces a general complex M-by-N matrix A to upper or lower
bidiagonal form B by a unitary transformation: Q**H * A * P = B.

If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.

Parameters

M

M is INTEGER
The number of rows in the matrix A. M >= 0.

N

N is INTEGER
The number of columns in the matrix A. N >= 0.

A

A is COMPLEX*16 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

LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).

D

D is DOUBLE PRECISION array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B:
D(i) = A(i,i).

E

E is DOUBLE PRECISION 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.

TAUQ

TAUQ is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.

TAUP

TAUP is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.

WORK

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is INTEGER
The length of the array WORK.
LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(M,N), otherwise.
For optimum performance LWORK >= (M+N)*NB, where NB
is the optimal blocksize.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.

INFO

INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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**H and G(i) = I - taup * u * u**H

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**H and G(i) = I - taup * u * u**H

where tauq and taup are complex scalars, and 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 )

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).

Author

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