LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ chpt21()

subroutine chpt21 ( integer  ITYPE,
character  UPLO,
integer  N,
integer  KBAND,
complex, dimension( * )  AP,
real, dimension( * )  D,
real, dimension( * )  E,
complex, dimension( ldu, * )  U,
integer  LDU,
complex, dimension( * )  VP,
complex, dimension( * )  TAU,
complex, dimension( * )  WORK,
real, dimension( * )  RWORK,
real, dimension( 2 )  RESULT 
)

CHPT21

Purpose:
 CHPT21  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
[in]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 )
[in]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.
[in]N
          N is INTEGER
          The size of the matrix.  If it is zero, CHPT21 does nothing.
          It must be at least zero.
[in]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.
[in]AP
          AP is COMPLEX 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.
[in]D
          D is REAL array, dimension (N)
          The diagonal of the (symmetric tri-) diagonal matrix.
[in]E
          E is REAL 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.
[in]U
          U is COMPLEX 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.
[in]LDU
          LDU is INTEGER
          The leading dimension of U.  LDU must be at least N and
          at least 1.
[in]VP
          VP is REAL 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.
[in]TAU
          TAU is COMPLEX 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.
[out]WORK
          WORK is COMPLEX array, dimension (N**2)
          Workspace.
[out]RWORK
          RWORK is REAL array, dimension (N)
          Workspace.
[out]RESULT
          RESULT is REAL 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 chpt21.f.

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  REAL D( * ), E( * ), RESULT( 2 ), RWORK( * )
239  COMPLEX AP( * ), TAU( * ), U( LDU, * ), VP( * ),
240  $ WORK( * )
241 * ..
242 *
243 * =====================================================================
244 *
245 * .. Parameters ..
246  REAL ZERO, ONE, TEN
247  parameter( zero = 0.0e+0, one = 1.0e+0, ten = 10.0e+0 )
248  REAL HALF
249  parameter( half = 1.0e+0 / 2.0e+0 )
250  COMPLEX CZERO, CONE
251  parameter( czero = ( 0.0e+0, 0.0e+0 ),
252  $ cone = ( 1.0e+0, 0.0e+0 ) )
253 * ..
254 * .. Local Scalars ..
255  LOGICAL LOWER
256  CHARACTER CUPLO
257  INTEGER IINFO, J, JP, JP1, JR, LAP
258  REAL ANORM, ULP, UNFL, WNORM
259  COMPLEX TEMP, VSAVE
260 * ..
261 * .. External Functions ..
262  LOGICAL LSAME
263  REAL CLANGE, CLANHP, SLAMCH
264  COMPLEX CDOTC
265  EXTERNAL lsame, clange, clanhp, slamch, cdotc
266 * ..
267 * .. External Subroutines ..
268  EXTERNAL caxpy, ccopy, cgemm, chpmv, chpr, chpr2,
269  $ clacpy, claset, cupmtr
270 * ..
271 * .. Intrinsic Functions ..
272  INTRINSIC cmplx, max, min, real
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 = slamch( 'Safe minimum' )
295  ulp = slamch( 'Epsilon' )*slamch( '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( clanhp( '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 claset( 'Full', n, n, czero, czero, work, n )
321  CALL ccopy( lap, ap, 1, work, 1 )
322 *
323  DO 10 j = 1, n
324  CALL chpr( 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 chpr2( cuplo, n, -cmplx( e( j ) ), u( 1, j ), 1,
330  $ u( 1, j-1 ), 1, work )
331  20 CONTINUE
332  END IF
333  wnorm = clanhp( '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 claset( '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 chpmv( 'L', n-j, cone, work( jp1+j+1 ),
357  $ vp( jp+j+1 ), 1, czero, work( lap+1 ), 1 )
358  temp = -half*tau( j )*cdotc( n-j, work( lap+1 ), 1,
359  $ vp( jp+j+1 ), 1 )
360  CALL caxpy( n-j, temp, vp( jp+j+1 ), 1, work( lap+1 ),
361  $ 1 )
362  CALL chpr2( '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 chpmv( 'U', j, cone, work, vp( jp1+1 ), 1, czero,
385  $ work( lap+1 ), 1 )
386  temp = -half*tau( j )*cdotc( j, work( lap+1 ), 1,
387  $ vp( jp1+1 ), 1 )
388  CALL caxpy( j, temp, vp( jp1+1 ), 1, work( lap+1 ),
389  $ 1 )
390  CALL chpr2( '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 = clanhp( '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 clacpy( ' ', n, n, u, ldu, work, n )
410  CALL cupmtr( '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 = clange( '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, real( 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 cgemm( '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( clange( '1', n, n, work, n, rwork ),
447  $ real( n ) ) / ( n*ulp )
448  END IF
449 *
450  RETURN
451 *
452 * End of CHPT21
453 *
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
complex function cdotc(N, CX, INCX, CY, INCY)
CDOTC
Definition: cdotc.f:83
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:88
subroutine chpr2(UPLO, N, ALPHA, X, INCX, Y, INCY, AP)
CHPR2
Definition: chpr2.f:145
subroutine chpr(UPLO, N, ALPHA, X, INCX, AP)
CHPR
Definition: chpr.f:130
subroutine chpmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
CHPMV
Definition: chpmv.f:149
subroutine cgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
CGEMM
Definition: cgemm.f:187
real function clange(NORM, M, N, A, LDA, WORK)
CLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: clange.f:115
subroutine claset(UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: claset.f:106
real function clanhp(NORM, UPLO, N, AP, WORK)
CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: clanhp.f:117
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
subroutine cupmtr(SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)
CUPMTR
Definition: cupmtr.f:150
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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