# zhet21.f - Man Page

TESTING/EIG/zhet21.f

## 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:**

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 'V' and whose scaling constants are in 'TAU'. We shall use the letter 'V' 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 ) 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.

## Referenced By

The man page zhet21(3) is an alias of zhet21.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK