 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ sspt21()

 subroutine sspt21 ( integer ITYPE, character UPLO, integer N, integer KBAND, real, dimension( * ) AP, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldu, * ) U, integer LDU, real, dimension( * ) VP, real, dimension( * ) TAU, real, dimension( * ) WORK, real, dimension( 2 ) RESULT )

SSPT21

Purpose:
``` SSPT21  generally checks a decomposition of the form

A = U S U**T

where **T means transpose, A is symmetric (stored in packed format), U
is orthogonal, 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) = | A - U S U**T | / ( |A| n ulp ) and
RESULT(2) = | I - U U**T | / ( n ulp )

If ITYPE=2, then:

RESULT(1) = | A - V S V**T | / ( |A| n ulp )

If ITYPE=3, then:

RESULT(1) = | I - V U**T | / ( 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)**T

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)**T

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 orthogonal matrix: RESULT(1) = | A - U S U**T | / ( |A| n ulp ) and RESULT(2) = | I - U U**T | / ( n ulp ) 2: U expressed as a product V of Housholder transformations: RESULT(1) = | A - V S V**T | / ( |A| n ulp ) 3: U expressed both as a dense orthogonal matrix and as a product of Housholder transformations: RESULT(1) = | I - V U**T | / ( n ulp )``` [in] UPLO ``` UPLO is CHARACTER If UPLO='U', AP and VP are considered to contain the upper triangle of A and V. If UPLO='L', AP and VP are considered to contain the lower triangle of A and V.``` [in] N ``` N is INTEGER The size of the matrix. If it is zero, SSPT21 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 REAL array, dimension (N*(N+1)/2) The original (unfactored) matrix. It is assumed to be symmetric, 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-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.``` [in] U ``` U is REAL array, dimension (LDU, N) If ITYPE=1 or 3, this contains the orthogonal 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 orthogonal 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 REAL array, dimension (N) If ITYPE >= 2, then TAU(j) is the scalar factor of v(j) v(j)**T 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 REAL array, dimension (N**2+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.```

Definition at line 219 of file sspt21.f.

221*
222* -- LAPACK test routine --
223* -- LAPACK is a software package provided by Univ. of Tennessee, --
224* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
225*
226* .. Scalar Arguments ..
227 CHARACTER UPLO
228 INTEGER ITYPE, KBAND, LDU, N
229* ..
230* .. Array Arguments ..
231 REAL AP( * ), D( * ), E( * ), RESULT( 2 ), TAU( * ),
232 \$ U( LDU, * ), VP( * ), WORK( * )
233* ..
234*
235* =====================================================================
236*
237* .. Parameters ..
238 REAL ZERO, ONE, TEN
239 parameter( zero = 0.0e0, one = 1.0e0, ten = 10.0e0 )
240 REAL HALF
241 parameter( half = 1.0e+0 / 2.0e+0 )
242* ..
243* .. Local Scalars ..
244 LOGICAL LOWER
245 CHARACTER CUPLO
246 INTEGER IINFO, J, JP, JP1, JR, LAP
247 REAL ANORM, TEMP, ULP, UNFL, VSAVE, WNORM
248* ..
249* .. External Functions ..
250 LOGICAL LSAME
251 REAL SDOT, SLAMCH, SLANGE, SLANSP
252 EXTERNAL lsame, sdot, slamch, slange, slansp
253* ..
254* .. External Subroutines ..
255 EXTERNAL saxpy, scopy, sgemm, slacpy, slaset, sopmtr,
256 \$ sspmv, sspr, sspr2
257* ..
258* .. Intrinsic Functions ..
259 INTRINSIC max, min, real
260* ..
261* .. Executable Statements ..
262*
263* 1) Constants
264*
265 result( 1 ) = zero
266 IF( itype.EQ.1 )
267 \$ result( 2 ) = zero
268 IF( n.LE.0 )
269 \$ RETURN
270*
271 lap = ( n*( n+1 ) ) / 2
272*
273 IF( lsame( uplo, 'U' ) ) THEN
274 lower = .false.
275 cuplo = 'U'
276 ELSE
277 lower = .true.
278 cuplo = 'L'
279 END IF
280*
281 unfl = slamch( 'Safe minimum' )
282 ulp = slamch( 'Epsilon' )*slamch( 'Base' )
283*
284* Some Error Checks
285*
286 IF( itype.LT.1 .OR. itype.GT.3 ) THEN
287 result( 1 ) = ten / ulp
288 RETURN
289 END IF
290*
291* Do Test 1
292*
293* Norm of A:
294*
295 IF( itype.EQ.3 ) THEN
296 anorm = one
297 ELSE
298 anorm = max( slansp( '1', cuplo, n, ap, work ), unfl )
299 END IF
300*
301* Compute error matrix:
302*
303 IF( itype.EQ.1 ) THEN
304*
305* ITYPE=1: error = A - U S U**T
306*
307 CALL slaset( 'Full', n, n, zero, zero, work, n )
308 CALL scopy( lap, ap, 1, work, 1 )
309*
310 DO 10 j = 1, n
311 CALL sspr( cuplo, n, -d( j ), u( 1, j ), 1, work )
312 10 CONTINUE
313*
314 IF( n.GT.1 .AND. kband.EQ.1 ) THEN
315 DO 20 j = 1, n - 1
316 CALL sspr2( cuplo, n, -e( j ), u( 1, j ), 1, u( 1, j+1 ),
317 \$ 1, work )
318 20 CONTINUE
319 END IF
320 wnorm = slansp( '1', cuplo, n, work, work( n**2+1 ) )
321*
322 ELSE IF( itype.EQ.2 ) THEN
323*
324* ITYPE=2: error = V S V**T - A
325*
326 CALL slaset( 'Full', n, n, zero, zero, work, n )
327*
328 IF( lower ) THEN
329 work( lap ) = d( n )
330 DO 40 j = n - 1, 1, -1
331 jp = ( ( 2*n-j )*( j-1 ) ) / 2
332 jp1 = jp + n - j
333 IF( kband.EQ.1 ) THEN
334 work( jp+j+1 ) = ( one-tau( j ) )*e( j )
335 DO 30 jr = j + 2, n
336 work( jp+jr ) = -tau( j )*e( j )*vp( jp+jr )
337 30 CONTINUE
338 END IF
339*
340 IF( tau( j ).NE.zero ) THEN
341 vsave = vp( jp+j+1 )
342 vp( jp+j+1 ) = one
343 CALL sspmv( 'L', n-j, one, work( jp1+j+1 ),
344 \$ vp( jp+j+1 ), 1, zero, work( lap+1 ), 1 )
345 temp = -half*tau( j )*sdot( n-j, work( lap+1 ), 1,
346 \$ vp( jp+j+1 ), 1 )
347 CALL saxpy( n-j, temp, vp( jp+j+1 ), 1, work( lap+1 ),
348 \$ 1 )
349 CALL sspr2( 'L', n-j, -tau( j ), vp( jp+j+1 ), 1,
350 \$ work( lap+1 ), 1, work( jp1+j+1 ) )
351 vp( jp+j+1 ) = vsave
352 END IF
353 work( jp+j ) = d( j )
354 40 CONTINUE
355 ELSE
356 work( 1 ) = d( 1 )
357 DO 60 j = 1, n - 1
358 jp = ( j*( j-1 ) ) / 2
359 jp1 = jp + j
360 IF( kband.EQ.1 ) THEN
361 work( jp1+j ) = ( one-tau( j ) )*e( j )
362 DO 50 jr = 1, j - 1
363 work( jp1+jr ) = -tau( j )*e( j )*vp( jp1+jr )
364 50 CONTINUE
365 END IF
366*
367 IF( tau( j ).NE.zero ) THEN
368 vsave = vp( jp1+j )
369 vp( jp1+j ) = one
370 CALL sspmv( 'U', j, one, work, vp( jp1+1 ), 1, zero,
371 \$ work( lap+1 ), 1 )
372 temp = -half*tau( j )*sdot( j, work( lap+1 ), 1,
373 \$ vp( jp1+1 ), 1 )
374 CALL saxpy( j, temp, vp( jp1+1 ), 1, work( lap+1 ),
375 \$ 1 )
376 CALL sspr2( 'U', j, -tau( j ), vp( jp1+1 ), 1,
377 \$ work( lap+1 ), 1, work )
378 vp( jp1+j ) = vsave
379 END IF
380 work( jp1+j+1 ) = d( j+1 )
381 60 CONTINUE
382 END IF
383*
384 DO 70 j = 1, lap
385 work( j ) = work( j ) - ap( j )
386 70 CONTINUE
387 wnorm = slansp( '1', cuplo, n, work, work( lap+1 ) )
388*
389 ELSE IF( itype.EQ.3 ) THEN
390*
391* ITYPE=3: error = U V**T - I
392*
393 IF( n.LT.2 )
394 \$ RETURN
395 CALL slacpy( ' ', n, n, u, ldu, work, n )
396 CALL sopmtr( 'R', cuplo, 'T', n, n, vp, tau, work, n,
397 \$ work( n**2+1 ), iinfo )
398 IF( iinfo.NE.0 ) THEN
399 result( 1 ) = ten / ulp
400 RETURN
401 END IF
402*
403 DO 80 j = 1, n
404 work( ( n+1 )*( j-1 )+1 ) = work( ( n+1 )*( j-1 )+1 ) - one
405 80 CONTINUE
406*
407 wnorm = slange( '1', n, n, work, n, work( n**2+1 ) )
408 END IF
409*
410 IF( anorm.GT.wnorm ) THEN
411 result( 1 ) = ( wnorm / anorm ) / ( n*ulp )
412 ELSE
413 IF( anorm.LT.one ) THEN
414 result( 1 ) = ( min( wnorm, n*anorm ) / anorm ) / ( n*ulp )
415 ELSE
416 result( 1 ) = min( wnorm / anorm, real( n ) ) / ( n*ulp )
417 END IF
418 END IF
419*
420* Do Test 2
421*
422* Compute U U**T - I
423*
424 IF( itype.EQ.1 ) THEN
425 CALL sgemm( 'N', 'C', n, n, n, one, u, ldu, u, ldu, zero, work,
426 \$ n )
427*
428 DO 90 j = 1, n
429 work( ( n+1 )*( j-1 )+1 ) = work( ( n+1 )*( j-1 )+1 ) - one
430 90 CONTINUE
431*
432 result( 2 ) = min( slange( '1', n, n, work, n,
433 \$ work( n**2+1 ) ), real( n ) ) / ( n*ulp )
434 END IF
435*
436 RETURN
437*
438* End of SSPT21
439*
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:114
real function slansp(NORM, UPLO, N, AP, WORK)
SLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slansp.f:114
subroutine sopmtr(SIDE, UPLO, TRANS, M, N, AP, TAU, C, LDC, WORK, INFO)
SOPMTR
Definition: sopmtr.f:150
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
real function sdot(N, SX, INCX, SY, INCY)
SDOT
Definition: sdot.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine sspr(UPLO, N, ALPHA, X, INCX, AP)
SSPR
Definition: sspr.f:127
subroutine sspmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
SSPMV
Definition: sspmv.f:147
subroutine sspr2(UPLO, N, ALPHA, X, INCX, Y, INCY, AP)
SSPR2
Definition: sspr2.f:142
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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