# zhpt21.f - Man Page

TESTING/EIG/zhpt21.f

## Synopsis

### Functions/Subroutines

subroutine **zhpt21** (itype, uplo, n, kband, ap, d, e, u, ldu, vp, tau, work, rwork, result)**ZHPT21**

## 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**

**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 '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 ) 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

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zhpt21(3) is an alias of zhpt21.f(3).

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