CHETD2(3)

💅 CHETD2(3) — Library Functions Manual 💅

okay bestie, welcome to CHETD2(3) from the Library Functions Manual. This is the group-chat explainer for the Fortran source file, and yes she has lore. 🎀

🦋 NAME 🦋

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/chetd2.f — that's the source file this page documents, periodt. No cap, that's literally the NAME section. 💅

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine CHETD2 (uplo, n, a, lda, d, e, tau, info)

CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm). That's the whole vibe. 🦋

🎀 Function/Subroutine Documentation 🎀

subroutine CHETD2 (character uplo, integer n, complex, dimension( lda, * )
a, integer lda, real, dimension( * ) d, real, dimension( * ) e, complex,
dimension( * ) tau, integer info)

CHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm). ✨

🌸 Purpose 🌸

The whole purpose, bestie: CHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation: Q**H * A * Q = T. That's it, that's the assignment. 💖

🧸 Parameters 🧸

• UPLO — UPLO is CHARACTER*1. 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. N >= 0.
• 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 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 unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over-written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, 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).
• D — D is REAL array, dimension (N). The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
• E — E is REAL 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 — TAU is COMPLEX array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details).
• INFO — 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**H

where tau is a complex scalar, and v is a complex 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**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+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 174 of file chetd2.f.

👑 Author 👑

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