ZLATRZ(3)

💖 ZLATRZ(3) 💖

okay so welcome to the Library Functions Manual moment for ZLATRZ, bestie. it’s giving “tiny subroutine, big unitary energy,” and we’re gonna keep every single detail because that’s the whole point. 🫶

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🌸 NAME 🌸

the file this page is all about is literally:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zlatrz.f

that’s the icon, the main character, the source path. periodt. 💅

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✨ SYNOPSIS ✨

🦋 Functions/Subroutines 🦋

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

the one-line tagline is: ZLATRZ factors an upper trapezoidal matrix by means of unitary transformations. no cap, that’s the whole vibe. ✨

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🦋 Function/Subroutine Documentation 🦋

here’s the full signature, exactly as it is, so nobody comes for me:

subroutine ZLATRZ (integer m, integer n, integer l, complex*16, dimension(
     lda, * ) a, integer lda, complex*16, dimension( * ) tau, complex*16,
     dimension( * ) work)

ZLATRZ factors an upper trapezoidal matrix by means of unitary transformations.

🔥 Purpose 🔥

okay so ZLATRZ factors the M-by-(M+L) complex upper trapezoidal matrix

[ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z by means
of unitary transformations, where  Z is an (M+L)-by-(M+L) unitary
matrix and, R and A1 are M-by-M upper triangular matrices.

that’s the whole purpose. Z is out here being an (M+L)-by-(M+L) unitary matrix, and R and A1 are M-by-M upper triangular matrices. iconic. 👑

🎀 Parameters 🎀

• M — M is INTEGER. it’s the number of rows of the matrix A. constraint: M >= 0.
• N — N is INTEGER. it’s the number of columns of the matrix A. constraint: N >= 0.
• L — L is INTEGER. it’s the number of columns of the matrix A containing the meaningful part of the Householder vectors. constraint: N-M >= L >= 0.
• A — A is COMPLEX*16 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 unitary matrix Z as a product of M elementary reflectors.
• LDA — LDA is INTEGER. it’s the leading dimension of the array A. constraint: LDA >= max(1,M).
• TAU — TAU is COMPLEX*16 array, dimension (M). these are the scalar factors of the elementary reflectors.
• WORK — WORK is COMPLEX*16 array, dimension (M). that’s it, that’s the work array.

🌟 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 so 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 )**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, 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 zlatrz.f.

🌟 Author 🌟

Generated automatically by Doxygen for LAPACK from the source code.

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later, bestie — footer tea is LAPACK Version 3.12.0 and the source/path is /home/abui...RC/zlatrz.f(3). there’s no separate date showing in this footer, so that’s the whole sign-off. 💅🎀