Man page - labrd(3)

Packages contains this manual

Manual

labrd

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine clabrd (integer m, integer n, integer nb, complex, dimension(lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * )e, complex, dimension( * ) tauq, complex, dimension( * ) taup, complex,dimension( ldx, * ) x, integer ldx, complex, dimension( ldy, * ) y,integer ldy)
subroutine dlabrd (integer m, integer n, integer nb, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( * ) d,double precision, dimension( * ) e, double precision, dimension( * )tauq, double precision, dimension( * ) taup, double precision,dimension( ldx, * ) x, integer ldx, double precision, dimension( ldy, *) y, integer ldy)
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)
subroutine zlabrd (integer m, integer n, integer nb, complex*16, dimension(lda, * ) a, integer lda, double precision, dimension( * ) d, doubleprecision, dimension( * ) e, complex*16, dimension( * ) tauq,complex*16, dimension( * ) taup, complex*16, dimension( ldx, * ) x,integer ldx, complex*16, dimension( ldy, * ) y, integer ldy)
Author

NAME

labrd - labrd: step in gebrd

SYNOPSIS

Functions

subroutine clabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine dlabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine slabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine zlabrd (m, n, nb, a, lda, d, e, tauq, taup, x, ldx, y, ldy)
ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Detailed Description

Function Documentation

subroutine clabrd (integer m, integer n, integer nb, complex, dimension(lda, * ) a, integer lda, real, dimension( * ) d, real, dimension( * )e, complex, dimension( * ) tauq, complex, dimension( * ) taup, complex,dimension( ldx, * ) x, integer ldx, complex, dimension( ldy, * ) y,integer ldy)

CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

CLABRD 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**H * 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 CGEBRD

Parameters

M

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

N

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

NB

NB is INTEGER
The number of leading rows and columns of A to be reduced.

A

A is COMPLEX 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

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

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).

E

E is REAL array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of
the reduced matrix.

TAUQ

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

TAUP

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

X

X is COMPLEX array, dimension (LDX,NB)
The m-by-nb matrix X required to update the unreduced part
of A.

LDX

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

Y

Y is COMPLEX array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part
of A.

LDY

LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).

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:

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

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**H 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**H - X*U**H.

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).

subroutine dlabrd (integer m, integer n, integer nb, double precision,dimension( lda, * ) a, integer lda, double precision, dimension( * ) d,double precision, dimension( * ) e, double precision, dimension( * )tauq, double precision, dimension( * ) taup, double precision,dimension( ldx, * ) x, integer ldx, double precision, dimension( ldy, *) y, integer ldy)

DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

DLABRD 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 DGEBRD

Parameters

M

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

N

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

NB

NB is INTEGER
The number of leading rows and columns of A to be reduced.

A

A is DOUBLE PRECISION 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.

LDA

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

D

D is 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

E is DOUBLE PRECISION array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of
the reduced matrix.

TAUQ

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

TAUP

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

X

X is DOUBLE PRECISION array, dimension (LDX,NB)
The m-by-nb matrix X required to update the unreduced part
of A.

LDX

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

Y

Y is DOUBLE PRECISION array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part
of A.

LDY

LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).

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:

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).

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.

Purpose:

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

Parameters

M

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

N

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

NB

NB is INTEGER
The number of leading rows and columns of A to be reduced.

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.

LDA

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

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).

E

E is REAL array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of
the reduced matrix.

TAUQ

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

TAUP

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

X

X is REAL array, dimension (LDX,NB)
The m-by-nb matrix X required to update the unreduced part
of A.

LDX

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

Y

Y is REAL array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part
of A.

LDY

LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).

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:

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).

subroutine zlabrd (integer m, integer n, integer nb, complex*16, dimension(lda, * ) a, integer lda, double precision, dimension( * ) d, doubleprecision, dimension( * ) e, complex*16, dimension( * ) tauq,complex*16, dimension( * ) taup, complex*16, dimension( ldx, * ) x,integer ldx, complex*16, dimension( ldy, * ) y, integer ldy)

ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.

Purpose:

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**H * 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

Parameters

M

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

N

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

NB

NB is INTEGER
The number of leading rows and columns of A to be reduced.

A

A is 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

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

D

D is 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

E is DOUBLE PRECISION array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of
the reduced matrix.

TAUQ

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

TAUP

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

X

X is COMPLEX*16 array, dimension (LDX,NB)
The m-by-nb matrix X required to update the unreduced part
of A.

LDX

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

Y

Y is COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part
of A.

LDY

LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).

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:

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

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**H 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**H - X*U**H.

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).

Author

Generated automatically by Doxygen for LAPACK from the source code.