latrz

slatrz

Functions

void slatrz(
    const INT           m,
    const INT           n,
    const INT           l,
          f32* restrict A,
    const INT           lda,
          f32* restrict tau,
          f32* restrict work
);

void slatrz(const INT m, const INT n, const INT l, f32 *restrict A, const INT lda, f32 *restrict tau, f32 *restrict work)

SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix [ A1 A2 ] = [ A(0:M-1,0:M-1) A(0:M-1,N-L:N-1) ] as ( R 0 ) * Z, by means of orthogonal transformations.

Z is an (M+L)-by-(M+L) orthogonal matrix and, R and A1 are M-by-M upper triangular matrices.

The factorization is obtained by Householder’s method. The kth transformation matrix, Z(k), which is used to introduce zeros into the (m-k)th row of A, is given in the form

Z(k) = ( I 0 ), ( 0 T(k) )

where

T(k) = I - tau*u(k)*u(k)**T, u(k) = ( 1 ), ( 0 ) ( z(k) )

tau is a scalar and z(k) is an l element vector. tau and z(k) are chosen to annihilate the elements of the kth row of A2.

The scalar tau is returned in the kth element of TAU and the vector u(k) in the kth row of A2, such that the elements of z(k) are in a(k, n-l), …, a(k, n-1). The elements of R are returned in the upper triangular part of A1.

Z is given by

Z = Z(0) * Z(1) * … * Z(m-1).

Parameters

in

m

The number of rows of the matrix A. m >= 0.

in

n

The number of columns of the matrix A. n >= 0.

in

l

The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= l >= 0.

inout

A

Double precision array, dimension (lda, n). On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factorized. On exit, the leading M-by-M upper triangular part of A contains the upper triangular matrix R, and elements N-L to N-1 of the first M rows of A, with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors.

in

lda

The leading dimension of the array A. lda >= max(1, m).

out

tau

Double precision array, dimension (m). The scalar factors of the elementary reflectors.

out

work

Double precision array, dimension (m).

dlatrz

Functions

void dlatrz(
    const INT           m,
    const INT           n,
    const INT           l,
          f64* restrict A,
    const INT           lda,
          f64* restrict tau,
          f64* restrict work
);

void dlatrz(const INT m, const INT n, const INT l, f64 *restrict A, const INT lda, f64 *restrict tau, f64 *restrict work)

DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix [ A1 A2 ] = [ A(0:M-1,0:M-1) A(0:M-1,N-L:N-1) ] as ( R 0 ) * Z, by means of orthogonal transformations.

Z is an (M+L)-by-(M+L) orthogonal matrix and, R and A1 are M-by-M upper triangular matrices.

The factorization is obtained by Householder’s method. The kth transformation matrix, Z(k), which is used to introduce zeros into the (m-k)th row of A, is given in the form

Z(k) = ( I 0 ), ( 0 T(k) )

where

T(k) = I - tau*u(k)*u(k)**T, u(k) = ( 1 ), ( 0 ) ( z(k) )

tau is a scalar and z(k) is an l element vector. tau and z(k) are chosen to annihilate the elements of the kth row of A2.

The scalar tau is returned in the kth element of TAU and the vector u(k) in the kth row of A2, such that the elements of z(k) are in a(k, n-l), …, a(k, n-1). The elements of R are returned in the upper triangular part of A1.

Z is given by

Z = Z(0) * Z(1) * … * Z(m-1).

Parameters

in

m

The number of rows of the matrix A. m >= 0.

in

n

The number of columns of the matrix A. n >= 0.

in

l

The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= l >= 0.

inout

A

Double precision array, dimension (lda, n). On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factorized. On exit, the leading M-by-M upper triangular part of A contains the upper triangular matrix R, and elements N-L to N-1 of the first M rows of A, with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors.

in

lda

The leading dimension of the array A. lda >= max(1, m).

out

tau

Double precision array, dimension (m). The scalar factors of the elementary reflectors.

out

work

Double precision array, dimension (m).

clatrz

Functions

void clatrz(
    const INT           m,
    const INT           n,
    const INT           l,
          c64* restrict A,
    const INT           lda,
          c64* restrict tau,
          c64* restrict work
);

void clatrz(const INT m, const INT n, const INT l, c64 *restrict A, const INT lda, c64 *restrict tau, c64 *restrict work)

CLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix [ A1 A2 ] = [ A(0:M-1,0:M-1) A(0:M-1,N-L:N-1) ] as ( R 0 ) * Z, by means of unitary transformations.

Z is an (M+L)-by-(M+L) unitary matrix and, R and A1 are M-by-M upper triangular matrices.

The factorization is obtained by Householder’s method. The kth transformation matrix, Z(k), which is used to introduce zeros into the (m-k)th row of A, is given in the form

Z(k) = ( I 0 ), ( 0 T(k) )

where

T(k) = I - tau*u(k)*u(k)**H, u(k) = ( 1 ), ( 0 ) ( z(k) )

tau is a scalar and z(k) is an l element vector. tau and z(k) are chosen to annihilate the elements of the kth row of A2.

The scalar tau is returned in the kth element of TAU and the vector u(k) in the kth row of A2, such that the elements of z(k) are in a(k, n-l), …, a(k, n-1). The elements of R are returned in the upper triangular part of A1.

Z is given by

Z = Z(0) * Z(1) * … * Z(m-1).

Parameters

in

m

The number of rows of the matrix A. m >= 0.

in

n

The number of columns of the matrix A. n >= 0.

in

l

The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= l >= 0.

inout

A

Complex array, dimension (lda, n). On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factorized. On exit, the leading M-by-M upper triangular part of A contains the upper triangular matrix R, and elements N-L to N-1 of the first M rows of A, with the array TAU, represent the unitary matrix Z as a product of M elementary reflectors.

in

lda

The leading dimension of the array A. lda >= max(1, m).

out

tau

Complex array, dimension (m). The scalar factors of the elementary reflectors.

out

work

Complex array, dimension (m).

zlatrz

Functions

void zlatrz(
    const INT            m,
    const INT            n,
    const INT            l,
          c128* restrict A,
    const INT            lda,
          c128* restrict tau,
          c128* restrict work
);

void zlatrz(const INT m, const INT n, const INT l, c128 *restrict A, const INT lda, c128 *restrict tau, c128 *restrict work)

ZLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix [ A1 A2 ] = [ A(0:M-1,0:M-1) A(0:M-1,N-L:N-1) ] as ( R 0 ) * Z, by means of unitary transformations.

Z is an (M+L)-by-(M+L) unitary matrix and, R and A1 are M-by-M upper triangular matrices.

The factorization is obtained by Householder’s method. The kth transformation matrix, Z(k), which is used to introduce zeros into the (m-k)th row of A, is given in the form

Z(k) = ( I 0 ), ( 0 T(k) )

where

T(k) = I - tau*u(k)*u(k)**H, u(k) = ( 1 ), ( 0 ) ( z(k) )

tau is a scalar and z(k) is an l element vector. tau and z(k) are chosen to annihilate the elements of the kth row of A2.

The scalar tau is returned in the kth element of TAU and the vector u(k) in the kth row of A2, such that the elements of z(k) are in a(k, n-l), …, a(k, n-1). The elements of R are returned in the upper triangular part of A1.

Z is given by

Z = Z(0) * Z(1) * … * Z(m-1).

Parameters

in

m

The number of rows of the matrix A. m >= 0.

in

n

The number of columns of the matrix A. n >= 0.

in

l

The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= l >= 0.

inout

A

Complex array, dimension (lda, n). On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factorized. On exit, the leading M-by-M upper triangular part of A contains the upper triangular matrix R, and elements N-L to N-1 of the first M rows of A, with the array TAU, represent the unitary matrix Z as a product of M elementary reflectors.

in

lda

The leading dimension of the array A. lda >= max(1, m).

out

tau

Complex array, dimension (m). The scalar factors of the elementary reflectors.

out

work

Complex array, dimension (m).