ZHPT21(3)

Welcome bestie 💖 you’ve found ZHPT21(3) in the Library Functions Manual, and it’s lowkey giving “LAPACK test routine that checks a Hermitian decomposition” energy. No cap. Let’s get into it ✨

💖 NAME 💖

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/zhpt21.f

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine ZHPT21 (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau,
     work, rwork, result)

ZHPT21 — main character behavior only 💅

🦋 Function/Subroutine Documentation 🦋

subroutine ZHPT21 (integer itype, character uplo, integer n, integer kband,
     complex*16, dimension( * ) ap, double precision, dimension( * ) d,
     double precision, dimension( * ) e, complex*16, dimension( ldu, * ) u,
     integer ldu, complex*16, dimension( * ) vp, complex*16, dimension( * )
     tau, complex*16, dimension( * ) work, double precision, dimension( * )
     rwork, double precision, dimension( 2 ) result)

ZHPT21

Okay so this whole page is about one subroutine, and it’s a test routine for a decomposition. Keep your eyes on the formulas, bestie — they’re exact and they matter 🫶

✨ Purpose ✨

ZHPT21 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). 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 — 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) = | 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 )

Packed storage means that, for example, if UPLO='U', then the columns of the upper triangle of A are stored one after another, so that A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if UPLO='L', then the columns of the lower triangle of A are stored one after another in AP, so that A(j+1,j+1) immediately follows A(n,j) in the array AP. This means that A(i,j) is stored in:

AP( i + j*(j-1)/2 )                 if UPLO='U'

AP( i + (2*n-j)*(j-1)/2 )           if UPLO='L'

The array VP bears the same relation to the matrix V that A does to AP.

For ITYPE > 1, the transformation U is expressed as a product of Householder transformations:

If UPLO='U', then V = H(n-1)...H(1), where

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

and the first j-1 elements of v(j) are stored in V(1:j-1,j+1), (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 )), the j-th element is 1, and the last n-j elements are 0.

If UPLO='L', then V = H(1)...H(n-1), where

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

and the first j elements of v(j) are 0, the (j+1)-st is 1, and the (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e., in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .)

🎀 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, ZHPT21 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.

AP

AP is COMPLEX*16 array, dimension (N*(N+1)/2). The original (unfactored) matrix. It is assumed to be hermitian, and contains the columns of just the upper triangle (UPLO='U') or only the lower triangle (UPLO='L'), packed one after another.

D

D is DOUBLE PRECISION array, dimension (N). The diagonal of the (symmetric tri-) diagonal matrix.

E

E is DOUBLE PRECISION array, dimension (N). 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.

VP

VP is DOUBLE PRECISION array, dimension (N*(N+1)/2). If ITYPE=2 or 3, the columns of this array contain the Householder vectors used to describe the unitary matrix in the decomposition, as described in purpose.

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.

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 (N**2). Workspace.

RWORK

RWORK is DOUBLE PRECISION array, dimension (N). Workspace.

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 226 of file zhpt21.f.

🌈 Author (yes, again) 🌈

Generated automatically by Doxygen for LAPACK from the source code.

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okay that’s the whole page, bestie 🫶 — signed off with LAPACK Version 3.12.0 and the footer path /home/abui...IG/zhpt21.f(3). Periodt 💅✨