TESTING/EIG/zhet21.f(3)

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version 3.12.1 · 3 versions
, in arch rolling

NAME 💖

TESTING/EIG/zhet21.f — that's the file, bestie. Hii and welcome to TESTING/EIG/zhet21.f(3) from the Library Functions Manual ✨

SYNOPSIS 🦋

Functions/Subroutines

subroutine zhet21 (itype, uplo, n, kband, a, lda, d, e, u, ldu, v, ldv,
           tau, work, rwork, result)

ZHET21

Function/Subroutine Documentation 📚

Here's the full signature, exactly as the page gives it:

subroutine zhet21 (integer itype, character uplo, integer n, 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)

ZHET21

Purpose 🌸

Okay so ZHET21 generally checks a decomposition of the form

A = U S U**H

where **H means conjugate transpose, A is hermitian, U is unitary, and S is diagonal (if KBAND=0) or (real) symmetric tridiagonal (if KBAND=1). Iconic behavior, honestly.

If ITYPE=1, then U is represented as a dense matrix; otherwise U is expressed as a product of Householder transformations, whose vectors are stored in the array and whose scaling constants are in . We shall use the letter to refer to the product of Householder transformations (which should be equal to U). (the source text has blanks there, and I'm leaving them exactly as-is 🫠)

Specifically, if ITYPE=1, then:

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

If ITYPE=2, then:

RESULT(1) = | A - V S V**H | / ( |A| n ulp )

If ITYPE=3, then:

RESULT(1) = | I - U V**H | / ( n ulp )

For ITYPE > 1, the transformation U is expressed as a product

V = H(1)...H(n-2)

where

H(j) = I  -  tau(j) v(j) v(j)**H

and each vector v(j) has its first j elements 0 and the remaining n-j elements stored in V(j+1:n,j). Slay 💅

Parameters 🎀

  • ITYPE — ITYPE is INTEGER. Specifies the type of tests to be performed.
    • 1: U expressed as a dense unitary matrix:
RESULT(1) = | A - U S U**H | / ( |A| n ulp ) and
RESULT(2) = | I - U U**H | / ( n ulp )
  • 2: U expressed as a product V of Housholder transformations:
RESULT(1) = | A - V S V**H | / ( |A| n ulp )
  • 3: U expressed both as a dense unitary matrix and as a product of Housholder transformations:
RESULT(1) = | I - U V**H | / ( n ulp )
  • UPLO — UPLO is CHARACTER. If UPLO='U', the upper triangle of A and V will be used and the (strictly) lower triangle will not be referenced. If UPLO='L', the lower triangle of A and V will be used and the (strictly) upper triangle will not be referenced.
  • N — N is INTEGER. The size of the matrix. If it is zero, ZHET21 does nothing. It must be at least zero.
  • KBAND — KBAND is 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.
  • A — A is COMPLEX*16 array, dimension (LDA, N). The original (unfactored) matrix. It is assumed to be hermitian, and only the upper (UPLO='U') or only the lower (UPLO='L') will be referenced.
  • LDA — LDA is INTEGER. The leading dimension of A. It must be at least 1 and at least N.
  • D — D is DOUBLE PRECISION array, dimension (N). The diagonal of the (symmetric tri-) diagonal matrix.
  • E — E is DOUBLE PRECISION array, dimension (N-1). The off-diagonal of the (symmetric tri-) diagonal matrix. E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and (3,2) element, etc. Not referenced if KBAND=0.
  • U — U is COMPLEX*16 array, dimension (LDU, N). If ITYPE=1 or 3, this contains the unitary matrix in the decomposition, expressed as a dense matrix. If ITYPE=2, then it is not referenced.
  • LDU — LDU is INTEGER. The leading dimension of U. LDU must be at least N and at least 1.
  • V — V is COMPLEX*16 array, dimension (LDV, N). If ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the unitary matrix in the decomposition. If UPLO='L', then the vectors are in the lower triangle, if UPLO='U', then in the upper triangle.
NOTE If ITYPE=2 or 3, V is modified and restored. The subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U') is set to one, and later reset to its original value, during the course of the calculation. If ITYPE=1, then it is neither referenced nor modified. so don't panic when it moves, it's coming back 🫶
  • LDV — LDV is INTEGER. The leading dimension of V. LDV must be at least N and at least 1.
  • TAU — TAU is 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.
  • WORK — WORK is COMPLEX*16 array, dimension (2*N**2).
  • RWORK — RWORK is DOUBLE PRECISION array, dimension (N).
  • RESULT — RESULT is 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 ITYPE=1.

Author 🌟

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 212 of file zhet21.f.

Author 🪄

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