ZLATRD(3)

💅 zlatrd.f(3) — ZLATRD 💅

Okay bestie, welcome to zlatrd.f(3) — a.k.a. ZLATRD — the manual page that’s giving big “LAPACK auxiliary routine with main character energy” vibes. The header calls itself Library Functions Manual and repeats the path /home/abui...RC/zlatrd.f(3) on both sides like it’s trying to be mysterious. The NAME section spills the whole address, and it is literally:

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

No cap, that’s the file. 🫶

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🎀 NAME 🎀

The NAME section is just that full path, bestie:

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

That’s it. That’s the name. Iconic, honestly. ✨

🦋 SYNOPSIS 🦋

Under Functions/Subroutines, the synopsis serves us this signature:

subroutine ZLATRD (uplo, n, nb, a, lda, e, tau, w, ldw)

And the one-liner is: ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation. Yes, “an unitary” — we’re keeping the original wording exactly, periodt.

🌸 Function/Subroutine Documentation 🌸

Bestie, here’s the full signature in all its detailed glory:

subroutine ZLATRD (character uplo, integer n, integer nb, complex*16,
    dimension( lda, * ) a, integer lda, double precision, dimension( * ) e,
    complex*16, dimension( * ) tau, complex*16, dimension( ldw, * ) w,
    integer ldw)

And the description again, because LAPACK said it twice and we respect the repetition: ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

🎯 Purpose 🎯

Okay so here’s the actual vibe of what it does:

ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to Hermitian tridiagonal form by a unitary similarity transformation Q**H * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by ZHETRD. 💅

🧚 Parameters 🧚

• UPLO — UPLO is CHARACTER*1. Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
  • = 'U': Upper triangular
  • = 'L': Lower triangular
• N — N is INTEGER. The order of the matrix A.
• NB — NB is INTEGER. The number of rows and columns to be reduced.
• A — A is COMPLEX*16 array, dimension (LDA,N).
  • On entry, the Hermitian matrix A. If UPLO = 'U', the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = 'L', the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
  • On exit: if UPLO = 'U', the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the unitary matrix Q as a product of elementary reflectors.
  • See Further Details.
• LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• E — E is DOUBLE PRECISION array, dimension (N-1). If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = 'L', E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix.
• TAU — TAU is COMPLEX*16 array, dimension (N-1). The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. See Further Details.
• W — W is COMPLEX*16 array, dimension (LDW,NB). The n-by-nb matrix W required to update the unreduced part of A.
• LDW — LDW is INTEGER. The leading dimension of the array W. LDW >= max(1,N).

👑 Author 👑

Respect to the author squad:

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

🌈 Further Details 🌈

If UPLO = 'U', the matrix Q is represented as a product of elementary reflectors:

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form:

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a Hermitian rank-2k update of the form:

A := A - V*W**H - W*V**H.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = 'U':                       if UPLO = 'L':

    (  a   a   a   v4  v5 )              (  d                  )
    (      a   a   v4  v5 )              (  1   d              )
    (          a   1   v5 )              (  v1  1   a          )
    (              d   1  )              (  v1  v2  a   a      )
    (                  d  )              (  v1  v2  a   a   a  )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i).

🧾 Definition 🧾

Definition at line 198 of file zlatrd.f.

👑 Author 👑

Generated automatically by Doxygen for LAPACK from the source code.

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Okay bestie, that’s the whole page — from LAPACK, Version 3.12.0, with the footer path /home/abui...RC/zlatrd.f(3). Periodt. 💅✨