SRC/clatrd.f(3)

💖 SRC/clatrd.f 💖

okay so bestie, this is the Library Functions Manual page for SRC/clatrd.f(3) — we folded the manual name right up here at the top because main character energy. 💅 This whole page is about clatrd, an auxiliary LAPACK routine that does tridiagonal reduction magic. 🪄

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✨ NAME ✨

SRC/clatrd.f

🌸 SYNOPSIS 🌸

Functions/Subroutines

• subroutine clatrd (uplo, n, nb, a, lda, e, tau, w, ldw) — CLATRD 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 clatrd signature 🎀

subroutine clatrd (character uplo, integer n, integer nb, complex,
    dimension( lda, * ) a, integer lda, real, dimension( * ) e, complex,
    dimension( * ) tau, complex, dimension( ldw, * ) w, integer ldw)

CLATRD 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 the purpose is: CLATRD 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', CLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = 'L', CLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied. 💅

This is an auxiliary routine called by CHETRD. 👑

🌈 Parameters 🌈

• UPLO — UPLO is CHARACTER*1; it 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. The size. The moment. 💫
• NB — NB is INTEGER; the number of rows and columns to be reduced.
• A — A is COMPLEX 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 REAL 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 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 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 👑

• 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, and it's giving:

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). Iconic. ✨

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 at line 198 of file clatrd.f.

🎀 Author 🎀

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💖 that's the whole page, bestie! 💖 source SRC/clatrd.f(3) • LAPACK Version 3.12.0 • the date slot isn't shown in this source footer, so no fake date energy 😭. Stay slay, and may your tridiagonal reductions be correct and your reflectors elementary. 🫶✨