SRC/dsytd2.f(3)

💖 NAME

SRC/dsytd2.f — yep, that’s the source file, bestie. And yo, this page is from the Library Functions Manual, so you know it’s giving official-library energy ✨

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✨ SYNOPSIS

🧚 Functions/Subroutines

• subroutine dsytd2 (uplo, n, a, lda, d, e, tau, info) — DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). 💅

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🫶 Function/Subroutine Documentation

subroutine dsytd2 (character uplo, integer n, double precision, dimension(
     lda, * ) a, integer lda, double precision, dimension( * ) d, double
     precision, dimension( * ) e, double precision, dimension( * ) tau,
     integer info)

DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). No cap, that’s the whole vibe of this subroutine 🦋

🎯 Purpose

!>
!> DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
!> form T by an orthogonal similarity transformation: Q**T * A * Q = T.
!>

So basically: symmetric matrix A walks in, tridiagonal form T walks out, and Q is the orthogonal similarity transformation doing the heavy lifting. Main character energy, honestly 👑

🌸 Parameters

• 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 — is INTEGER. The order of the matrix A. N >= 0.
• A — is DOUBLE PRECISION array, dimension (LDA,N). On entry, this is 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 diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. See Further Details.
• LDA — is INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• D — is DOUBLE PRECISION array, dimension (N). The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
• E — is DOUBLE PRECISION array, dimension (N-1). The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
• TAU — is DOUBLE PRECISION array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details).
• INFO — is INTEGER.
  • = 0: successful exit
  • < 0: if INFO = -i, the i-th argument had an illegal value.

👑 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-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in A(1:i-1,i+1), and tau in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary reflectors:

Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), and tau in TAU(i).

The contents of A on exit are illustrated by the following examples with n = 5:

if UPLO = 'U':                       if UPLO = 'L':

  (  d   e   v2  v3  v4 )              (  d                  )
  (      d   e   v3  v4 )              (  e   d              )
  (          d   e   v4 )              (  v1  e   d          )
  (              d   e  )              (  v1  v2  e   d      )
  (                  d  )              (  v1  v2  v3  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i).

Definition at line 172 of file dsytd2.f.

🎀 Author

Generated automatically by Doxygen for LAPACK from the source code.

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💖 okay bestie, that’s the whole page — brought to you by LAPACK Version 3.12.0, from the source file SRC/dsytd2.f(3), with all the tridiagonal details intact and zero facts dropped, periodt ✨🎀