CLATRD(3)

💅 CLATRD(3) — Library Functions Manual 💅

🎀 NAME 🎀

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/clatrd.f — okay so bestie, that’s the exact source file path, no cap. 🫶

✨ 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. Periodt. 💖

🦋 Function/Subroutine Documentation 🦋

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 🌸

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. That’s the whole assignment. 💜

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. Iykyk.

🍓 Parameters 🍓

• UPLO — CHARACTER*1. Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular; = 'L': Lower triangular. 💅
• N — INTEGER. The order of the matrix A. 🌈
• NB — INTEGER. The number of rows and columns to be reduced. ✨
• A — 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,N). 🎀
• E — 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 — 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 — COMPLEX array, dimension (LDW,NB). The n-by-nb matrix W required to update the unreduced part of A. 🦋
• LDW — 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.

Slay. 💅

🌈 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 at line 198 of file clatrd.f. 📌

👑 Author (Doxygen edition) 👑

Generated automatically by Doxygen for LAPACK from the source code. 🤖✨

💌 Cute sign-off: this footer energy is LAPACK Version 3.12.0 and /home/abui...RC/clatrd.f(3) — source/version noted, no separate date printed, and that’s the whole vibe. Mwah. 💋