SRC/zhetd2.f(3)

💅 SRC/zhetd2.f(3) — Library Functions Manual 💅

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✨ NAME ✨

SRC/zhetd2.f

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🌸 SYNOPSIS 🌸

✨ Functions/Subroutines ✨

subroutine zhetd2 (uplo, n, a, lda, d, e, tau, info)
    ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form
    by an unitary similarity transformation (unblocked algorithm).

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💖 Function/Subroutine Documentation 💖

   subroutine zhetd2 (character uplo, integer n, complex*16, dimension( lda, *
       ) a, integer lda, double precision, dimension( * ) d, double precision,
       dimension( * ) e, complex*16, dimension( * ) tau, integer info)

ZHETD2 reduces a Hermitian matrix to real symmetric tridiagonal form by an unitary similarity transformation (unblocked algorithm). 💫

🎀 Purpose 🎀

Vibe check: ZHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal form T by a unitary similarity transformation:

Q**H * A * Q = T.

🧚 Parameters 🧚

• UPLO — is CHARACTER*1 and specifies whether the upper or lower triangular part of the Hermitian matrix A is stored:
  • = 'U': Upper triangular
  • = 'L': Lower triangular
• N — is INTEGER. The order of the matrix A. N >= 0.
• A — is COMPLEX*16 array, dimension (LDA,N). On entry, it's 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. 👀
• 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 COMPLEX*16 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**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 zhetd2.f.

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🌟 Author 🌟

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