SRC/chetrd.f(3)

💖 SRC/chetrd.f 💖

okay bestie, this page is from the Library Functions Manual and it’s all about SRC/chetrd.f — the LAPACK function/subroutine doc for chetrd, aka CHETRD. main character energy, honestly.

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✨ SYNOPSIS ✨

Functions/Subroutines

• subroutine chetrd (uplo, n, a, lda, d, e, tau, work, lwork, info) — CHETRD

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📚 Function/Subroutine Documentation

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

CHETRD — the whole name of the game. periodt.

🎯 Purpose

okay so, CHETRD reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation:

Q**H * A * Q = T.

that’s the glow-up, bestie. ✨

🧩 Parameters

• UPLO — CHARACTER*1. 'U': upper triangle of A is stored; 'L': lower triangle of A is stored. tiny switch, big consequences.
• N — INTEGER. The order of the matrix A. N >= 0.
• 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 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 overwritten 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. she contains multitudes, fr fr.
• LDA — INTEGER. The leading dimension of the array A. LDA >= max(1,N).
• D — REAL array, dimension (N). The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).
• E — 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 — COMPLEX array, dimension (N-1). The scalar factors of the elementary reflectors (see Further Details).
• WORK — COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — INTEGER. The dimension of the array WORK. LWORK >= 1. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. workspace query said “no thoughts, just optimal size.”
• INFO — 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.

the whole academic-and-NAG friend group showed up for this one 💖

🔍 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). look at that little matrix art, she ate 💅

🧾 Definition at line 191 of file chetrd.f

🪄 Author

Generated automatically by Doxygen for LAPACK from the source code.

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okay bestie, footer vibe check: LAPACK, Version 3.12.0, source SRC/chetrd.f(3) — that’s the whole sign-off. bye babes ✨🫶