SRC/dlatrz.f(3)

This is the LAPACK manual page for SRC/dlatrz.f(3) from the Library Functions Manual — and yes bestie, it's giving "small but powerful linear algebra icon." 💅

✨ NAME ✨

SRC/dlatrz.f — that's the page name. The main character living here is dlatrz, and it's all about factoring an upper trapezoidal matrix with orthogonal transformations. 🫶

🌈 SYNOPSIS 🌈

Functions/Subroutines

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

🦋 Function/Subroutine Documentation 🦋

subroutine dlatrz (integer m, integer n, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) tau, double precision, dimension( * ) work)

DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations. Periodt. 💫

🎀 Purpose 🎀

Okay so here's the tea: DLATRZ 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, and it does this 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. No cap. ✨

💖 Parameters 💖

M

M is INTEGER. It's the number of rows of matrix A, and M >= 0. Slay. 🌸

N

N is INTEGER. It's the number of columns of matrix A, and N >= 0.

L

L is INTEGER. It's the number of columns of matrix A containing the meaningful part of the Householder vectors. Constraint check: N-M >= L >= 0.

A

A is 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+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. It's the leading dimension of the array A, and LDA >= max(1,M).

TAU

TAU is DOUBLE PRECISION array, dimension (M). It holds the scalar factors of the elementary reflectors. Iconic. 💅

WORK

WORK is DOUBLE PRECISION array, dimension (M). That's it. It works. 🫶

👑 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 🔥

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 dlatrz.f.

💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code.

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Cute sign-off, bestie: LAPACK Version 3.12.0 · SRC/dlatrz.f(3). The footer date isn't in the original, so I'm not inventing one — just version vibes and a whole lot of Householder energy. 💖✨