SRC/zlatrd.f(3)

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version 3.12.1 · 4 versions
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💅 zlatrd.f — SRC/zlatrd.f(3)

okay so bestie, sit down, because this one is a LAPACK internal routine and it's lowkey iconic. it lives in SRC/zlatrd.f, and it does exactly one job extremely well. let's get into it 🫶

✨ SYNOPSIS ✨

🌸 Functions/Subroutines

  • subroutine zlatrd (uplo, n, nb, a, lda, e, tau, w, ldw) — ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

🎀 Function/Subroutine Documentation 🎀

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)

same vibe as above: ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation. she's an overachiever, we stan 💜

🔥 Purpose 🔥

okay so here's the whole point of her existence:

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.

translation: she's the bestie that ZHETRD calls when it needs the job done. no cap. ✨

🦋 Parameters 🦋

  • UPLO — UPLO is CHARACTER*1. tells you 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. like, how many of them are getting the glow-up.
  • 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). don't try to be cute with this one, the bound is the bound. 💀
  • 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. her little helper. 🍓
  • LDW — LDW is INTEGER. the leading dimension of the array W. LDW >= max(1,N). again: respect the bound, bestie. 🙈

👑 Author 👑

  • 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':

  (  a   a   a   v4  v5 )
  (      a   a   v4  v5 )
  (          a   1   v5 )
  (              d   1  )
  (                  d  )

if UPLO = 'L':

  (  d                  )
  (  1   d              )
  (  v1  1   a          )
  (  v1  v2  a   a      )
  (  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). literally the whole glow-up documented in ASCII. obsessed. ✨💅

📍 Definition at line 198 of file zlatrd.f.

💖 Author 💖

Generated automatically by Doxygen for LAPACK from the source code.

and that's the whole page, bestie 🎀 you now know ZLATRD intimately. go forth and reduce some Hermitian matrices. 🦋

LAPACK — Version 3.12.0 — SRC/zlatrd.f(3) ✨💜

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