ZHET21(3)

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/zhet21.f — that’s the file this page is serving, bestie. Welcome to ZHET21(3) from the LAPACK 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 💖

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) 💅

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). (yes, the source text literally leaves those spots blank — I’m not making it up 🙈)

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).

🧚 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. 💀

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 🌈

Generated automatically by Doxygen for LAPACK from the source code.

---

Bye bestie 🫶 this icon of a page is from LAPACK Version 3.12.0, and the footer points at /home/abui...IG/zhet21.f(3). Slay, periodt. 💖✨