SRC/dlatrd.f(3)

✨ SRC/dlatrd.f(3) ✨

Library Functions Manual — okay bestie, welcome to the manual page vibe check 💅

🌸 NAME 🌸

SRC/dlatrd.f — that's the name, and we're already slaying. ✨

🦋 SYNOPSIS 🦋

💅 Functions/Subroutines 💅

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

DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation. 🫶

👑 Function/Subroutine Documentation 👑

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

DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation. No cap. 💖

🌟 Purpose 🌟

okay so purpose check: DLATRD takes NB rows and columns of a real symmetric matrix A and reduces them to symmetric tridiagonal form using the orthogonal similarity transformation Q**T * A * Q — and it hands back the matrices V and W that are needed to apply that transformation to the unreduced part of A. 💅

If UPLO = 'U', it reduces the last NB rows and columns of a matrix whose upper triangle is supplied. If UPLO = 'L', it reduces the first NB rows and columns of a matrix whose lower triangle is supplied. It's an auxiliary routine called by DSYTRD. Periodt. ✨

🎀 Parameters 🎀

💖 UPLO 💖

UPLO is CHARACTER*1. It specifies whether the upper or lower triangular part of the symmetric 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 DOUBLE PRECISION array, dimension (LDA,N). On entry, it's the symmetric 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 orthogonal 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 orthogonal matrix Q as a product of elementary reflectors.

See Further Details. 🦋

💖 LDA 💖

LDA is INTEGER. The leading dimension of the array A. LDA >= (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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 👑

Shoutout to the authors, absolute icons:

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

🔥 Further Details 🔥

okay, further details time, and yes this is the part with the reflector lore 🧚

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**T

where tau is a real scalar, and v is a real 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**T

where tau is a real scalar, and v is a real 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 — together with W — to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form:

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

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 197 of file dlatrd.f. 📍

👑 Author 👑

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

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mwah 💋 — this was LAPACK Version 3.12.0 on page SRC/dlatrd.f(3), and you ate that, bestie. 🫶✨