DLATRZ(3)

🌸 DLATRZ(3) — the girlies' guide to LAPACK 🌸

Okay so bestie, first things first: this page lives at /home/abui...RC/dlatrz.f(3) in the Library Functions Manual. That's the whole vibe. Now let's get into it, because this subroutine is literally iconic. 💅

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✨ SYNOPSIS ✨

Functions/Subroutines

subroutine DLATRZ (m, n, l, a, lda, tau, work)

DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations. No cap, that's the whole main character energy of this routine. 🎀

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🦋 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)

Okay so here's the tea: DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations. Full stop. Periodt. 💜

🔥 Purpose:

!> 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, 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.

Like, it takes this matrix and just rearranges it with orthogonal transformations, and R and A1 come out as M-by-M upper triangular. It's giving structure. It's giving triangles. Obsessed. 🌈

💖 Parameters

• M — bestie, this is an INTEGER. It's the number of rows of the matrix A. Requirement: M >= 0. Don't forget, okay?

!>          M is INTEGER
!>          The number of rows of the matrix A.  M >= 0.

• N — also an INTEGER, iconic. The number of columns of the matrix A. Requirement: N >= 0. Keep those non-negative, queen.

!>          N is INTEGER
!>          The number of columns of the matrix A.  N >= 0.

• L — another INTEGER, we love consistency. The number of columns of the matrix A containing the meaningful part of the Householder vectors. Requirement: N-M >= L >= 0. Read that carefully, bestie — it's a two-sided inequality, no skipping. 💀

!>          L is INTEGER
!>          The number of columns of the matrix A containing the
!>          meaningful part of the Householder vectors. N-M >= L >= 0.

• 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+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. It goes in trapezoidal and comes out triangular, a true glow-up. ✨

!>          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 — INTEGER, and it's the leading dimension of the array A. Requirement: LDA >= max(1,M). Don't let LDA be too small, she needs room to breathe. 🫶

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

• TAU — DOUBLE PRECISION array, dimension (M). The scalar factors of the elementary reflectors. These are the little scalars that make the magic happen.

!>          TAU is DOUBLE PRECISION array, dimension (M)
!>          The scalar factors of the elementary reflectors.

• WORK — DOUBLE PRECISION array, dimension (M). That's it. That's the whole description. She's lowkey mysterious like that. 🙈

!>          WORK is DOUBLE PRECISION array, dimension (M)

👑 Author

Shoutout to the absolute legends who made this possible:

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

Collab of the century, honestly. 🌈

🧚 Contributors:

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

Okay A. Petitet, we see you, and we appreciate you. 💖

⭐ 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 fortran !> Z( k ) = ( I 0 ), !> ( 0 T( k ) ) !> where fortran !> 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 fortran !> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).

Okay so the math is giving Householder realness — it uses these elementary reflectors to annihilate stuff and build Z as a product of all the Z( k ) factors. It's literally a chain of transformations and it slays. 🔥

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📌 Just so you know, this whole thing is defined at line 139 of file dlatrz.f.

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🎀 Author (of this fine manual page)

Generated automatically by Doxygen for LAPACK from the source code.

Not me being auto-generated and still serving looks. 💅

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✨ That's the whole vibe check, bestie! ✨ This page is from LAPACK Version 3.12.0 and lives at /home/abui...RC/dlatrz.f(3). Stay slay, stay orthogonal, and remember: M >= 0, N >= 0, N-M >= L >= 0, and LDA >= max(1,M). Periodt. 💖🔥🦋