Man page - hptrd(3)

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Manual

hptrd

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine chptrd (character uplo, integer n, complex, dimension( * ) ap,real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * )tau, integer info)
subroutine dsptrd (character uplo, integer n, double precision, dimension(* ) ap, double precision, dimension( * ) d, double precision,dimension( * ) e, double precision, dimension( * ) tau, integer info)
subroutine ssptrd (character uplo, integer n, real, dimension( * ) ap,real, dimension( * ) d, real, dimension( * ) e, real, dimension( * )tau, integer info)
subroutine zhptrd (character uplo, integer n, complex*16, dimension( * )ap, double precision, dimension( * ) d, double precision, dimension( *) e, complex*16, dimension( * ) tau, integer info)
Author

NAME

hptrd - {hp,sp}trd: reduction to tridiagonal

SYNOPSIS

Functions

subroutine chptrd (uplo, n, ap, d, e, tau, info)
CHPTRD
subroutine dsptrd (uplo, n, ap, d, e, tau, info)
DSPTRD
subroutine ssptrd (uplo, n, ap, d, e, tau, info)
SSPTRD
subroutine zhptrd (uplo, n, ap, d, e, tau, info)
ZHPTRD

Detailed Description

Function Documentation

subroutine chptrd (character uplo, integer n, complex, dimension( * ) ap,real, dimension( * ) d, real, dimension( * ) e, complex, dimension( * )tau, integer info)

CHPTRD

Purpose:

CHPTRD reduces a complex Hermitian matrix A stored in packed form to
real symmetric tridiagonal form T by a unitary similarity
transformation: Q**H * A * Q = T.

Parameters

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

N is INTEGER
The order of the matrix A. N >= 0.

AP

AP is COMPLEX array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = ā€™Uā€™, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ā€™Lā€™, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, if UPLO = ā€™Uā€™, the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the unitary
matrix Q as a product of elementary reflectors; if UPLO
= ā€™Lā€™, the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the unitary matrix Q as a product
of elementary reflectors. See Further Details.

D

D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).

E

E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = ā€™Uā€™, E(i) = A(i+1,i) if UPLO = ā€™Lā€™.

TAU

TAU is COMPLEX array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

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:

If UPLO = ā€™Uā€™, the matrix Q is represented as a product of elementary
reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = ā€™Lā€™, the matrix Q is represented as a product of elementary
reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).

subroutine dsptrd (character uplo, integer n, double precision, dimension(* ) ap, double precision, dimension( * ) d, double precision,dimension( * ) e, double precision, dimension( * ) tau, integer info)

DSPTRD

Purpose:

DSPTRD reduces a real symmetric matrix A stored in packed form to
symmetric tridiagonal form T by an orthogonal similarity
transformation: Q**T * A * Q = T.

Parameters

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

N is INTEGER
The order of the matrix A. N >= 0.

AP

AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = ā€™Uā€™, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ā€™Lā€™, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, if UPLO = ā€™Uā€™, the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= ā€™Lā€™, the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.

D

D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).

E

E is DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = ā€™Uā€™, E(i) = A(i+1,i) if UPLO = ā€™Lā€™.

TAU

TAU is DOUBLE PRECISION array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

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:

If UPLO = ā€™Uā€™, the matrix Q is represented as a product of elementary
reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = ā€™Lā€™, the matrix Q is represented as a product of elementary
reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).

subroutine ssptrd (character uplo, integer n, real, dimension( * ) ap,real, dimension( * ) d, real, dimension( * ) e, real, dimension( * )tau, integer info)

SSPTRD

Purpose:

SSPTRD reduces a real symmetric matrix A stored in packed form to
symmetric tridiagonal form T by an orthogonal similarity
transformation: Q**T * A * Q = T.

Parameters

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

N is INTEGER
The order of the matrix A. N >= 0.

AP

AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = ā€™Uā€™, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ā€™Lā€™, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, if UPLO = ā€™Uā€™, the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the orthogonal
matrix Q as a product of elementary reflectors; if UPLO
= ā€™Lā€™, the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the orthogonal matrix Q as a product
of elementary reflectors. See Further Details.

D

D is REAL array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).

E

E is REAL array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = ā€™Uā€™, E(i) = A(i+1,i) if UPLO = ā€™Lā€™.

TAU

TAU is REAL array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

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:

If UPLO = ā€™Uā€™, the matrix Q is represented as a product of elementary
reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = ā€™Lā€™, the matrix Q is represented as a product of elementary
reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).

subroutine zhptrd (character uplo, integer n, complex*16, dimension( * )ap, double precision, dimension( * ) d, double precision, dimension( *) e, complex*16, dimension( * ) tau, integer info)

ZHPTRD

Purpose:

ZHPTRD reduces a complex Hermitian matrix A stored in packed form to
real symmetric tridiagonal form T by a unitary similarity
transformation: Q**H * A * Q = T.

Parameters

UPLO

UPLO is CHARACTER*1
= ā€™Uā€™: Upper triangle of A is stored;
= ā€™Lā€™: Lower triangle of A is stored.

N

N is INTEGER
The order of the matrix A. N >= 0.

AP

AP is COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = ā€™Uā€™, AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = ā€™Lā€™, AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
On exit, if UPLO = ā€™Uā€™, the diagonal and first superdiagonal
of A are overwritten by the corresponding elements of the
tridiagonal matrix T, and the elements above the first
superdiagonal, with the array TAU, represent the unitary
matrix Q as a product of elementary reflectors; if UPLO
= ā€™Lā€™, the diagonal and first subdiagonal of A are over-
written by the corresponding elements of the tridiagonal
matrix T, and the elements below the first subdiagonal, with
the array TAU, represent the unitary matrix Q as a product
of elementary reflectors. See Further Details.

D

D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of the tridiagonal matrix T:
D(i) = A(i,i).

E

E is DOUBLE PRECISION array, dimension (N-1)
The off-diagonal elements of the tridiagonal matrix T:
E(i) = A(i,i+1) if UPLO = ā€™Uā€™, E(i) = A(i+1,i) if UPLO = ā€™Lā€™.

TAU

TAU is COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors (see Further
Details).

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:

If UPLO = ā€™Uā€™, the matrix Q is represented as a product of elementary
reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = ā€™Lā€™, the matrix Q is represented as a product of elementary
reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).

Author

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