 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ zhet22()

 subroutine zhet22 ( integer ITYPE, character UPLO, integer N, integer M, 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 )

ZHET22

Purpose:
```      ZHET22  generally checks a decomposition of the form

A U = U S

where A is complex Hermitian, the columns of U are orthonormal,
and S is diagonal (if KBAND=0) or 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) = | U**H A U - S | / ( |A| m ulp ) and
RESULT(2) = | I - U**H U | / ( m ulp )```
```  ITYPE   INTEGER
Specifies the type of tests to be performed.
1: U expressed as a dense orthogonal matrix:
RESULT(1) = | A - U S U**H | / ( |A| n ulp )   *and
RESULT(2) = | I - U U**H | / ( n ulp )

UPLO    CHARACTER
If UPLO='U', the upper triangle of A will be used and the
(strictly) lower triangle will not be referenced.  If
UPLO='L', the lower triangle of A will be used and the
(strictly) upper triangle will not be referenced.
Not modified.

N       INTEGER
The size of the matrix.  If it is zero, ZHET22 does nothing.
It must be at least zero.
Not modified.

M       INTEGER
The number of columns of U.  If it is zero, ZHET22 does
nothing.  It must be at least zero.
Not modified.

KBAND   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.
Not modified.

A       COMPLEX*16 array, dimension (LDA , N)
The original (unfactored) matrix.  It is assumed to be
symmetric, and only the upper (UPLO='U') or only the lower
(UPLO='L') will be referenced.
Not modified.

LDA     INTEGER
The leading dimension of A.  It must be at least 1
and at least N.
Not modified.

D       DOUBLE PRECISION array, dimension (N)
The diagonal of the (symmetric tri-) diagonal matrix.
Not modified.

E       DOUBLE PRECISION array, dimension (N)
The off-diagonal of the (symmetric tri-) diagonal matrix.
E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
Not referenced if KBAND=0.
Not modified.

U       COMPLEX*16 array, dimension (LDU, N)
If ITYPE=1, this contains the orthogonal matrix in
the decomposition, expressed as a dense matrix.
Not modified.

LDU     INTEGER
The leading dimension of U.  LDU must be at least N and
at least 1.
Not modified.

V       COMPLEX*16 array, dimension (LDV, N)
If ITYPE=2 or 3, the lower triangle of this array contains
the Householder vectors used to describe the orthogonal
matrix in the decomposition.  If ITYPE=1, then it is not
referenced.
Not modified.

LDV     INTEGER
The leading dimension of V.  LDV must be at least N and
at least 1.
Not modified.

TAU     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.
Not modified.

WORK    COMPLEX*16 array, dimension (2*N**2)
Workspace.
Modified.

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

RESULT  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 LDU is at least N.
Modified.```

Definition at line 159 of file zhet22.f.

161*
162* -- LAPACK test routine --
163* -- LAPACK is a software package provided by Univ. of Tennessee, --
164* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
165*
166* .. Scalar Arguments ..
167 CHARACTER UPLO
168 INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
169* ..
170* .. Array Arguments ..
171 DOUBLE PRECISION D( * ), E( * ), RESULT( 2 ), RWORK( * )
172 COMPLEX*16 A( LDA, * ), TAU( * ), U( LDU, * ),
173 \$ V( LDV, * ), WORK( * )
174* ..
175*
176* =====================================================================
177*
178* .. Parameters ..
179 DOUBLE PRECISION ZERO, ONE
180 parameter( zero = 0.0d0, one = 1.0d0 )
181 COMPLEX*16 CZERO, CONE
182 parameter( czero = ( 0.0d0, 0.0d0 ),
183 \$ cone = ( 1.0d0, 0.0d0 ) )
184* ..
185* .. Local Scalars ..
186 INTEGER J, JJ, JJ1, JJ2, NN, NNP1
187 DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
188* ..
189* .. External Functions ..
190 DOUBLE PRECISION DLAMCH, ZLANHE
191 EXTERNAL dlamch, zlanhe
192* ..
193* .. External Subroutines ..
194 EXTERNAL zgemm, zhemm, zunt01
195* ..
196* .. Intrinsic Functions ..
197 INTRINSIC dble, max, min
198* ..
199* .. Executable Statements ..
200*
201 result( 1 ) = zero
202 result( 2 ) = zero
203 IF( n.LE.0 .OR. m.LE.0 )
204 \$ RETURN
205*
206 unfl = dlamch( 'Safe minimum' )
207 ulp = dlamch( 'Precision' )
208*
209* Do Test 1
210*
211* Norm of A:
212*
213 anorm = max( zlanhe( '1', uplo, n, a, lda, rwork ), unfl )
214*
215* Compute error matrix:
216*
217* ITYPE=1: error = U**H A U - S
218*
219 CALL zhemm( 'L', uplo, n, m, cone, a, lda, u, ldu, czero, work,
220 \$ n )
221 nn = n*n
222 nnp1 = nn + 1
223 CALL zgemm( 'C', 'N', m, m, n, cone, u, ldu, work, n, czero,
224 \$ work( nnp1 ), n )
225 DO 10 j = 1, m
226 jj = nn + ( j-1 )*n + j
227 work( jj ) = work( jj ) - d( j )
228 10 CONTINUE
229 IF( kband.EQ.1 .AND. n.GT.1 ) THEN
230 DO 20 j = 2, m
231 jj1 = nn + ( j-1 )*n + j - 1
232 jj2 = nn + ( j-2 )*n + j
233 work( jj1 ) = work( jj1 ) - e( j-1 )
234 work( jj2 ) = work( jj2 ) - e( j-1 )
235 20 CONTINUE
236 END IF
237 wnorm = zlanhe( '1', uplo, m, work( nnp1 ), n, rwork )
238*
239 IF( anorm.GT.wnorm ) THEN
240 result( 1 ) = ( wnorm / anorm ) / ( m*ulp )
241 ELSE
242 IF( anorm.LT.one ) THEN
243 result( 1 ) = ( min( wnorm, m*anorm ) / anorm ) / ( m*ulp )
244 ELSE
245 result( 1 ) = min( wnorm / anorm, dble( m ) ) / ( m*ulp )
246 END IF
247 END IF
248*
249* Do Test 2
250*
251* Compute U**H U - I
252*
253 IF( itype.EQ.1 )
254 \$ CALL zunt01( 'Columns', n, m, u, ldu, work, 2*n*n, rwork,
255 \$ result( 2 ) )
256*
257 RETURN
258*
259* End of ZHET22
260*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine zhemm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZHEMM
Definition: zhemm.f:191
subroutine zunt01(ROWCOL, M, N, U, LDU, WORK, LWORK, RWORK, RESID)
ZUNT01
Definition: zunt01.f:126
double precision function zlanhe(NORM, UPLO, N, A, LDA, WORK)
ZLANHE returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: zlanhe.f:124
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