SRC/slatrz.f(3)

NAME 🎀

SRC/slatrz.f

okay so bestie, this page is from the Library Functions Manual and it's all about SRC/slatrz.f(3) — yes, that's the header vibe, folded in cute. no cap. 💖

SYNOPSIS ✨

Functions/Subroutines

• subroutine slatrz (m, n, l, a, lda, tau, work) — SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.

Function/Subroutine Documentation 🎀

subroutine slatrz (integer m, integer n, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( * ) tau, real, dimension( * ) work)

SLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.

Purpose 🌸

okay so here's the purpose, literally: SLATRZ factors the M-by-(M+L) real upper trapezoidal matrix [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] 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. periodt. 💅

Parameters 🦋

vibe check, one by one:

• M — M is INTEGER. The number of rows of the matrix A. M >= 0.
• N — N is INTEGER. The number of columns of the matrix A. N >= 0.
• L — L is INTEGER. The number of columns of the matrix A containing the meaningful part of the Householder vectors. N-M >= L >= 0.
• A — A is REAL 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+1 to N of the first M rows of A, with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• TAU — TAU is REAL array, dimension (M). The scalar factors of the elementary reflectors.
• WORK — WORK is REAL array, dimension (M).

Author 💖

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

Contributors 👑

• A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Further Details 🔥

okay this part is lowkey iconic: the factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the ( m - k + 1 )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, l + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of A1.

Z is given by

Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).

Definition at line 139 of file slatrz.f.

Author 🫶

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