SRC/slatrd.f(3)

💅 SRC/slatrd.f(3) — Library Functions Manual ✨

🎀 NAME

SRC/slatrd.f

okay so this is the page for SRC/slatrd.f — the slatrd subroutine. she’s iconic, she’s auxiliary, and she’s about to reduce some matrix rows and columns. 💖

✨ SYNOPSIS

🧚 Functions/Subroutines

• subroutine slatrd (uplo, n, nb, a, lda, e, tau, w, ldw) — SLATRD 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 slatrd (character uplo, integer n, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, real, dimension( * ) tau, real, dimension( ldw, * ) w, integer ldw)

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

🦋 Purpose:

okay bestie, here’s the actual lore: SLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q**T * 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', SLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied.
• if UPLO = 'L', SLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by SSYTRD. periodt.

🍓 Parameters

• UPLO — UPLO is CHARACTER*1. 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 REAL array, dimension (LDA,N). On entry, 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 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 REAL 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 REAL 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:

okay, this is the part where the reflectors get their moment. 💅

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, 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

Definition at line 197 of file slatrd.f.

🌟 Author

Generated automatically by Doxygen for LAPACK from the source code.

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💖 That’s the whole page, bestie. It’s LAPACK Version 3.12.0, source file SRC/slatrd.f(3) — go slay your tridiagonal reduction. 💅✨