LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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zhpt21.f
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1*> \brief \b ZHPT21
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8* Definition:
9* ===========
10*
11* SUBROUTINE ZHPT21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
12* TAU, WORK, RWORK, RESULT )
13*
14* .. Scalar Arguments ..
15* CHARACTER UPLO
16* INTEGER ITYPE, KBAND, LDU, N
17* ..
18* .. Array Arguments ..
19* DOUBLE PRECISION D( * ), E( * ), RESULT( 2 ), RWORK( * )
20* COMPLEX*16 AP( * ), TAU( * ), U( LDU, * ), VP( * ),
21* $ WORK( * )
22* ..
23*
24*
25*> \par Purpose:
26* =============
27*>
28*> \verbatim
29*>
30*> ZHPT21 generally checks a decomposition of the form
31*>
32*> A = U S U**H
33*>
34*> where **H means conjugate transpose, A is hermitian, U is
35*> unitary, and S is diagonal (if KBAND=0) or (real) symmetric
36*> tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as
37*> a dense matrix, otherwise the U is expressed as a product of
38*> Householder transformations, whose vectors are stored in the
39*> array "V" and whose scaling constants are in "TAU"; we shall
40*> use the letter "V" to refer to the product of Householder
41*> transformations (which should be equal to U).
42*>
43*> Specifically, if ITYPE=1, then:
44*>
45*> RESULT(1) = | A - U S U**H | / ( |A| n ulp ) and
46*> RESULT(2) = | I - U U**H | / ( n ulp )
47*>
48*> If ITYPE=2, then:
49*>
50*> RESULT(1) = | A - V S V**H | / ( |A| n ulp )
51*>
52*> If ITYPE=3, then:
53*>
54*> RESULT(1) = | I - U V**H | / ( n ulp )
55*>
56*> Packed storage means that, for example, if UPLO='U', then the columns
57*> of the upper triangle of A are stored one after another, so that
58*> A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if
59*> UPLO='L', then the columns of the lower triangle of A are stored one
60*> after another in AP, so that A(j+1,j+1) immediately follows A(n,j)
61*> in the array AP. This means that A(i,j) is stored in:
62*>
63*> AP( i + j*(j-1)/2 ) if UPLO='U'
64*>
65*> AP( i + (2*n-j)*(j-1)/2 ) if UPLO='L'
66*>
67*> The array VP bears the same relation to the matrix V that A does to
68*> AP.
69*>
70*> For ITYPE > 1, the transformation U is expressed as a product
71*> of Householder transformations:
72*>
73*> If UPLO='U', then V = H(n-1)...H(1), where
74*>
75*> H(j) = I - tau(j) v(j) v(j)**H
76*>
77*> and the first j-1 elements of v(j) are stored in V(1:j-1,j+1),
78*> (i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ),
79*> the j-th element is 1, and the last n-j elements are 0.
80*>
81*> If UPLO='L', then V = H(1)...H(n-1), where
82*>
83*> H(j) = I - tau(j) v(j) v(j)**H
84*>
85*> and the first j elements of v(j) are 0, the (j+1)-st is 1, and the
86*> (j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e.,
87*> in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .)
88*> \endverbatim
89*
90* Arguments:
91* ==========
92*
93*> \param[in] ITYPE
94*> \verbatim
95*> ITYPE is INTEGER
96*> Specifies the type of tests to be performed.
97*> 1: U expressed as a dense unitary matrix:
98*> RESULT(1) = | A - U S U**H | / ( |A| n ulp ) and
99*> RESULT(2) = | I - U U**H | / ( n ulp )
100*>
101*> 2: U expressed as a product V of Housholder transformations:
102*> RESULT(1) = | A - V S V**H | / ( |A| n ulp )
103*>
104*> 3: U expressed both as a dense unitary matrix and
105*> as a product of Housholder transformations:
106*> RESULT(1) = | I - U V**H | / ( n ulp )
107*> \endverbatim
108*>
109*> \param[in] UPLO
110*> \verbatim
111*> UPLO is CHARACTER
112*> If UPLO='U', the upper triangle of A and V will be used and
113*> the (strictly) lower triangle will not be referenced.
114*> If UPLO='L', the lower triangle of A and V will be used and
115*> the (strictly) upper triangle will not be referenced.
116*> \endverbatim
117*>
118*> \param[in] N
119*> \verbatim
120*> N is INTEGER
121*> The size of the matrix. If it is zero, ZHPT21 does nothing.
122*> It must be at least zero.
123*> \endverbatim
124*>
125*> \param[in] KBAND
126*> \verbatim
127*> KBAND is INTEGER
128*> The bandwidth of the matrix. It may only be zero or one.
129*> If zero, then S is diagonal, and E is not referenced. If
130*> one, then S is symmetric tri-diagonal.
131*> \endverbatim
132*>
133*> \param[in] AP
134*> \verbatim
135*> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
136*> The original (unfactored) matrix. It is assumed to be
137*> hermitian, and contains the columns of just the upper
138*> triangle (UPLO='U') or only the lower triangle (UPLO='L'),
139*> packed one after another.
140*> \endverbatim
141*>
142*> \param[in] D
143*> \verbatim
144*> D is DOUBLE PRECISION array, dimension (N)
145*> The diagonal of the (symmetric tri-) diagonal matrix.
146*> \endverbatim
147*>
148*> \param[in] E
149*> \verbatim
150*> E is DOUBLE PRECISION array, dimension (N)
151*> The off-diagonal of the (symmetric tri-) diagonal matrix.
152*> E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
153*> (3,2) element, etc.
154*> Not referenced if KBAND=0.
155*> \endverbatim
156*>
157*> \param[in] U
158*> \verbatim
159*> U is COMPLEX*16 array, dimension (LDU, N)
160*> If ITYPE=1 or 3, this contains the unitary matrix in
161*> the decomposition, expressed as a dense matrix. If ITYPE=2,
162*> then it is not referenced.
163*> \endverbatim
164*>
165*> \param[in] LDU
166*> \verbatim
167*> LDU is INTEGER
168*> The leading dimension of U. LDU must be at least N and
169*> at least 1.
170*> \endverbatim
171*>
172*> \param[in] VP
173*> \verbatim
174*> VP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
175*> If ITYPE=2 or 3, the columns of this array contain the
176*> Householder vectors used to describe the unitary matrix
177*> in the decomposition, as described in purpose.
178*> *NOTE* If ITYPE=2 or 3, V is modified and restored. The
179*> subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
180*> is set to one, and later reset to its original value, during
181*> the course of the calculation.
182*> If ITYPE=1, then it is neither referenced nor modified.
183*> \endverbatim
184*>
185*> \param[in] TAU
186*> \verbatim
187*> TAU is COMPLEX*16 array, dimension (N)
188*> If ITYPE >= 2, then TAU(j) is the scalar factor of
189*> v(j) v(j)**H in the Householder transformation H(j) of
190*> the product U = H(1)...H(n-2)
191*> If ITYPE < 2, then TAU is not referenced.
192*> \endverbatim
193*>
194*> \param[out] WORK
195*> \verbatim
196*> WORK is COMPLEX*16 array, dimension (N**2)
197*> Workspace.
198*> \endverbatim
199*>
200*> \param[out] RWORK
201*> \verbatim
202*> RWORK is DOUBLE PRECISION array, dimension (N)
203*> Workspace.
204*> \endverbatim
205*>
206*> \param[out] RESULT
207*> \verbatim
208*> RESULT is DOUBLE PRECISION array, dimension (2)
209*> The values computed by the two tests described above. The
210*> values are currently limited to 1/ulp, to avoid overflow.
211*> RESULT(1) is always modified. RESULT(2) is modified only
212*> if ITYPE=1.
213*> \endverbatim
214*
215* Authors:
216* ========
217*
218*> \author Univ. of Tennessee
219*> \author Univ. of California Berkeley
220*> \author Univ. of Colorado Denver
221*> \author NAG Ltd.
222*
223*> \ingroup complex16_eig
224*
225* =====================================================================
226 SUBROUTINE zhpt21( ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP,
227 $ TAU, WORK, RWORK, RESULT )
228*
229* -- LAPACK test routine --
230* -- LAPACK is a software package provided by Univ. of Tennessee, --
231* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
232*
233* .. Scalar Arguments ..
234 CHARACTER UPLO
235 INTEGER ITYPE, KBAND, LDU, N
236* ..
237* .. Array Arguments ..
238 DOUBLE PRECISION D( * ), E( * ), RESULT( 2 ), RWORK( * )
239 COMPLEX*16 AP( * ), TAU( * ), U( LDU, * ), VP( * ),
240 $ work( * )
241* ..
242*
243* =====================================================================
244*
245* .. Parameters ..
246 DOUBLE PRECISION ZERO, ONE, TEN
247 parameter( zero = 0.0d+0, one = 1.0d+0, ten = 10.0d+0 )
248 DOUBLE PRECISION HALF
249 parameter( half = 1.0d+0 / 2.0d+0 )
250 COMPLEX*16 CZERO, CONE
251 parameter( czero = ( 0.0d+0, 0.0d+0 ),
252 $ cone = ( 1.0d+0, 0.0d+0 ) )
253* ..
254* .. Local Scalars ..
255 LOGICAL LOWER
256 CHARACTER CUPLO
257 INTEGER IINFO, J, JP, JP1, JR, LAP
258 DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
259 COMPLEX*16 TEMP, VSAVE
260* ..
261* .. External Functions ..
262 LOGICAL LSAME
263 DOUBLE PRECISION DLAMCH, ZLANGE, ZLANHP
264 COMPLEX*16 ZDOTC
265 EXTERNAL lsame, dlamch, zlange, zlanhp, zdotc
266* ..
267* .. External Subroutines ..
268 EXTERNAL zaxpy, zcopy, zgemm, zhpmv, zhpr, zhpr2,
270* ..
271* .. Intrinsic Functions ..
272 INTRINSIC dble, dcmplx, max, min
273* ..
274* .. Executable Statements ..
275*
276* Constants
277*
278 result( 1 ) = zero
279 IF( itype.EQ.1 )
280 $ result( 2 ) = zero
281 IF( n.LE.0 )
282 $ RETURN
283*
284 lap = ( n*( n+1 ) ) / 2
285*
286 IF( lsame( uplo, 'U' ) ) THEN
287 lower = .false.
288 cuplo = 'U'
289 ELSE
290 lower = .true.
291 cuplo = 'L'
292 END IF
293*
294 unfl = dlamch( 'Safe minimum' )
295 ulp = dlamch( 'Epsilon' )*dlamch( 'Base' )
296*
297* Some Error Checks
298*
299 IF( itype.LT.1 .OR. itype.GT.3 ) THEN
300 result( 1 ) = ten / ulp
301 RETURN
302 END IF
303*
304* Do Test 1
305*
306* Norm of A:
307*
308 IF( itype.EQ.3 ) THEN
309 anorm = one
310 ELSE
311 anorm = max( zlanhp( '1', cuplo, n, ap, rwork ), unfl )
312 END IF
313*
314* Compute error matrix:
315*
316 IF( itype.EQ.1 ) THEN
317*
318* ITYPE=1: error = A - U S U**H
319*
320 CALL zlaset( 'Full', n, n, czero, czero, work, n )
321 CALL zcopy( lap, ap, 1, work, 1 )
322*
323 DO 10 j = 1, n
324 CALL zhpr( cuplo, n, -d( j ), u( 1, j ), 1, work )
325 10 CONTINUE
326*
327 IF( n.GT.1 .AND. kband.EQ.1 ) THEN
328 DO 20 j = 2, n - 1
329 CALL zhpr2( cuplo, n, -dcmplx( e( j ) ), u( 1, j ), 1,
330 $ u( 1, j-1 ), 1, work )
331 20 CONTINUE
332 END IF
333 wnorm = zlanhp( '1', cuplo, n, work, rwork )
334*
335 ELSE IF( itype.EQ.2 ) THEN
336*
337* ITYPE=2: error = V S V**H - A
338*
339 CALL zlaset( 'Full', n, n, czero, czero, work, n )
340*
341 IF( lower ) THEN
342 work( lap ) = d( n )
343 DO 40 j = n - 1, 1, -1
344 jp = ( ( 2*n-j )*( j-1 ) ) / 2
345 jp1 = jp + n - j
346 IF( kband.EQ.1 ) THEN
347 work( jp+j+1 ) = ( cone-tau( j ) )*e( j )
348 DO 30 jr = j + 2, n
349 work( jp+jr ) = -tau( j )*e( j )*vp( jp+jr )
350 30 CONTINUE
351 END IF
352*
353 IF( tau( j ).NE.czero ) THEN
354 vsave = vp( jp+j+1 )
355 vp( jp+j+1 ) = cone
356 CALL zhpmv( 'L', n-j, cone, work( jp1+j+1 ),
357 $ vp( jp+j+1 ), 1, czero, work( lap+1 ), 1 )
358 temp = -half*tau( j )*zdotc( n-j, work( lap+1 ), 1,
359 $ vp( jp+j+1 ), 1 )
360 CALL zaxpy( n-j, temp, vp( jp+j+1 ), 1, work( lap+1 ),
361 $ 1 )
362 CALL zhpr2( 'L', n-j, -tau( j ), vp( jp+j+1 ), 1,
363 $ work( lap+1 ), 1, work( jp1+j+1 ) )
364*
365 vp( jp+j+1 ) = vsave
366 END IF
367 work( jp+j ) = d( j )
368 40 CONTINUE
369 ELSE
370 work( 1 ) = d( 1 )
371 DO 60 j = 1, n - 1
372 jp = ( j*( j-1 ) ) / 2
373 jp1 = jp + j
374 IF( kband.EQ.1 ) THEN
375 work( jp1+j ) = ( cone-tau( j ) )*e( j )
376 DO 50 jr = 1, j - 1
377 work( jp1+jr ) = -tau( j )*e( j )*vp( jp1+jr )
378 50 CONTINUE
379 END IF
380*
381 IF( tau( j ).NE.czero ) THEN
382 vsave = vp( jp1+j )
383 vp( jp1+j ) = cone
384 CALL zhpmv( 'U', j, cone, work, vp( jp1+1 ), 1, czero,
385 $ work( lap+1 ), 1 )
386 temp = -half*tau( j )*zdotc( j, work( lap+1 ), 1,
387 $ vp( jp1+1 ), 1 )
388 CALL zaxpy( j, temp, vp( jp1+1 ), 1, work( lap+1 ),
389 $ 1 )
390 CALL zhpr2( 'U', j, -tau( j ), vp( jp1+1 ), 1,
391 $ work( lap+1 ), 1, work )
392 vp( jp1+j ) = vsave
393 END IF
394 work( jp1+j+1 ) = d( j+1 )
395 60 CONTINUE
396 END IF
397*
398 DO 70 j = 1, lap
399 work( j ) = work( j ) - ap( j )
400 70 CONTINUE
401 wnorm = zlanhp( '1', cuplo, n, work, rwork )
402*
403 ELSE IF( itype.EQ.3 ) THEN
404*
405* ITYPE=3: error = U V**H - I
406*
407 IF( n.LT.2 )
408 $ RETURN
409 CALL zlacpy( ' ', n, n, u, ldu, work, n )
410 CALL zupmtr( 'R', cuplo, 'C', n, n, vp, tau, work, n,
411 $ work( n**2+1 ), iinfo )
412 IF( iinfo.NE.0 ) THEN
413 result( 1 ) = ten / ulp
414 RETURN
415 END IF
416*
417 DO 80 j = 1, n
418 work( ( n+1 )*( j-1 )+1 ) = work( ( n+1 )*( j-1 )+1 ) - cone
419 80 CONTINUE
420*
421 wnorm = zlange( '1', n, n, work, n, rwork )
422 END IF
423*
424 IF( anorm.GT.wnorm ) THEN
425 result( 1 ) = ( wnorm / anorm ) / ( n*ulp )
426 ELSE
427 IF( anorm.LT.one ) THEN
428 result( 1 ) = ( min( wnorm, n*anorm ) / anorm ) / ( n*ulp )
429 ELSE
430 result( 1 ) = min( wnorm / anorm, dble( n ) ) / ( n*ulp )
431 END IF
432 END IF
433*
434* Do Test 2
435*
436* Compute U U**H - I
437*
438 IF( itype.EQ.1 ) THEN
439 CALL zgemm( 'N', 'C', n, n, n, cone, u, ldu, u, ldu, czero,
440 $ work, n )
441*
442 DO 90 j = 1, n
443 work( ( n+1 )*( j-1 )+1 ) = work( ( n+1 )*( j-1 )+1 ) - cone
444 90 CONTINUE
445*
446 result( 2 ) = min( zlange( '1', n, n, work, n, rwork ),
447 $ dble( n ) ) / ( n*ulp )
448 END IF
449*
450 RETURN
451*
452* End of ZHPT21
453*
454 END
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zhpr(UPLO, N, ALPHA, X, INCX, AP)
ZHPR
Definition: zhpr.f:130
subroutine zhpr2(UPLO, N, ALPHA, X, INCX, Y, INCY, AP)
ZHPR2
Definition: zhpr2.f:145
subroutine zhpmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
ZHPMV
Definition: zhpmv.f:149
subroutine zgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZGEMM
Definition: zgemm.f:187
subroutine zhpt21(ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, TAU, WORK, RWORK, RESULT)
ZHPT21
Definition: zhpt21.f:228
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlaset(UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: zlaset.f:106
subroutine zupmtr(SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)
ZUPMTR
Definition: zupmtr.f:150