TESTING/EIG/zhet22.f(3)

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🎀 TESTING/EIG/zhet22.f(3) 🎀

okay bestie, welcome to the Library Functions Manual page for TESTING/EIG/zhet22.f(3). Buckle up, because this one is a whole little LAPACK test subroutine and we're gonna get into every single detail. 💅✨

🌸 NAME 🌸

TESTING/EIG/zhet22.f — that's the name, bestie. The whole page is literally this file. Periodt. 💖

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine zhet22 (itype, uplo, n, m, kband, a, lda, d, e, u, ldu, v,
           ldv, tau, work, rwork, result)
           ZHET22

🦋 Function/Subroutine Documentation 🦋

ZHET22

subroutine zhet22 (integer itype, character uplo, integer n, integer m,
       integer kband, complex*16, dimension( lda, * ) a, integer lda, double
       precision, dimension( * ) d, double precision, dimension( * ) e,
       complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension(
       ldv, * ) v, integer ldv, complex*16, dimension( * ) tau, complex*16,
       dimension( * ) work, double precision, dimension( * ) rwork, double
       precision, dimension( 2 ) result)
       ZHET22

Purpose

okay so ZHET22 generally checks a decomposition of the form

A U = U S

where A is complex Hermitian, the columns of U are orthonormal, and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as a dense matrix, otherwise the U is expressed as a product of Householder transformations, whose vectors are stored in the array and whose scaling constants are in — and then the original says, literally, “we shall use the letter” to refer to the product of Householder transformations (which should be equal to U). 🫶

Specifically, if ITYPE=1, then:

RESULT(1) = | U**H A U - S | / ( |A| m ulp ) and
RESULT(2) = | I - U**H U | / ( m ulp )

okay bestie, argument time — every single one, no skipping:

  • ITYPE INTEGER — Specifies the type of tests to be performed. 1: U expressed as a dense orthogonal matrix:
RESULT(1) = | A - U S U**H | / ( |A| n ulp )   *and
RESULT(2) = | I - U U**H | / ( n ulp )

(Yes, the original has *and — iconic, and I’m not fixing it. 😭)

  • UPLO CHARACTER — If UPLO='U', the upper triangle of A will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A will be used and the (strictly) upper triangle will not be referenced. Not modified.
  • N INTEGER — The size of the matrix. If it is zero, ZHET22 does nothing. It must be at least zero. Not modified.
  • M INTEGER — The number of columns of U. If it is zero, ZHET22 does nothing. It must be at least zero. Not modified.
  • KBAND INTEGER — The bandwidth of the matrix. It may only be zero or one. If zero, then S is diagonal, and E is not referenced. If one, then S is symmetric tri-diagonal. Not modified.
  • A COMPLEX*16 array, dimension (LDA , N) — The original (unfactored) matrix. It is assumed to be symmetric, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced. Not modified.
  • LDA INTEGER — The leading dimension of A. It must be at least 1 and at least N. Not modified.
  • D DOUBLE PRECISION array, dimension (N) — The diagonal of the (symmetric tri-) diagonal matrix. Not modified.
  • E DOUBLE PRECISION array, dimension (N) — The off-diagonal of the (symmetric tri-) diagonal matrix. E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc. Not referenced if KBAND=0. Not modified.
  • U COMPLEX*16 array, dimension (LDU, N) — If ITYPE=1, this contains the orthogonal matrix in the decomposition, expressed as a dense matrix. Not modified.
  • LDU INTEGER — The leading dimension of U. LDU must be at least N and at least 1. Not modified.
  • V COMPLEX*16 array, dimension (LDV, N) — If ITYPE=2 or 3, the lower triangle of this array contains the Householder vectors used to describe the orthogonal matrix in the decomposition. If ITYPE=1, then it is not referenced. Not modified.
  • LDV INTEGER — The leading dimension of V. LDV must be at least N and at least 1. Not modified.
  • TAU COMPLEX*16 array, dimension (N) — If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)**H in the Householder transformation H(j) of the product U = H(1)...H(n-2). If ITYPE < 2, then TAU is not referenced. Not modified.
  • WORK COMPLEX*16 array, dimension (2*N**2) — Workspace. Modified.
  • RWORK DOUBLE PRECISION array, dimension (N) — Workspace. Modified.
  • RESULT DOUBLE PRECISION array, dimension (2) — The values computed by the two tests described above. The values are currently limited to 1/ulp, to avoid overflow. RESULT(1) is always modified. RESULT(2) is modified only if LDU is at least N. Modified.

Author

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 159 of file zhet22.f.

🫶 Author 🫶

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