LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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◆ dtprfs()

subroutine dtprfs ( character  uplo,
character  trans,
character  diag,
integer  n,
integer  nrhs,
double precision, dimension( * )  ap,
double precision, dimension( ldb, * )  b,
integer  ldb,
double precision, dimension( ldx, * )  x,
integer  ldx,
double precision, dimension( * )  ferr,
double precision, dimension( * )  berr,
double precision, dimension( * )  work,
integer, dimension( * )  iwork,
integer  info 
)

DTPRFS

Download DTPRFS + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DTPRFS provides error bounds and backward error estimates for the
 solution to a system of linear equations with a triangular packed
 coefficient matrix.

 The solution matrix X must be computed by DTPTRS or some other
 means before entering this routine.  DTPRFS does not do iterative
 refinement because doing so cannot improve the backward error.
Parameters
[in]UPLO
          UPLO is CHARACTER*1
          = 'U':  A is upper triangular;
          = 'L':  A is lower triangular.
[in]TRANS
          TRANS is CHARACTER*1
          Specifies the form of the system of equations:
          = 'N':  A * X = B  (No transpose)
          = 'T':  A**T * X = B  (Transpose)
          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
[in]DIAG
          DIAG is CHARACTER*1
          = 'N':  A is non-unit triangular;
          = 'U':  A is unit triangular.
[in]N
          N is INTEGER
          The order of the matrix A.  N >= 0.
[in]NRHS
          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.
[in]AP
          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in
          a linear array.  The j-th column of A is stored in the array
          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
          If DIAG = 'U', the diagonal elements of A are not referenced
          and are assumed to be 1.
[in]B
          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
          The right hand side matrix B.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
[in]X
          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
          The solution matrix X.
[in]LDX
          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).
[out]FERR
          FERR is DOUBLE PRECISION array, dimension (NRHS)
          The estimated forward error bound for each solution vector
          X(j) (the j-th column of the solution matrix X).
          If XTRUE is the true solution corresponding to X(j), FERR(j)
          is an estimated upper bound for the magnitude of the largest
          element in (X(j) - XTRUE) divided by the magnitude of the
          largest element in X(j).  The estimate is as reliable as
          the estimate for RCOND, and is almost always a slight
          overestimate of the true error.
[out]BERR
          BERR is DOUBLE PRECISION array, dimension (NRHS)
          The componentwise relative backward error of each solution
          vector X(j) (i.e., the smallest relative change in
          any element of A or B that makes X(j) an exact solution).
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (3*N)
[out]IWORK
          IWORK is INTEGER array, dimension (N)
[out]INFO
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 173 of file dtprfs.f.

175*
176* -- LAPACK computational routine --
177* -- LAPACK is a software package provided by Univ. of Tennessee, --
178* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
179*
180* .. Scalar Arguments ..
181 CHARACTER DIAG, TRANS, UPLO
182 INTEGER INFO, LDB, LDX, N, NRHS
183* ..
184* .. Array Arguments ..
185 INTEGER IWORK( * )
186 DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
187 $ WORK( * ), X( LDX, * )
188* ..
189*
190* =====================================================================
191*
192* .. Parameters ..
193 DOUBLE PRECISION ZERO
194 parameter( zero = 0.0d+0 )
195 DOUBLE PRECISION ONE
196 parameter( one = 1.0d+0 )
197* ..
198* .. Local Scalars ..
199 LOGICAL NOTRAN, NOUNIT, UPPER
200 CHARACTER TRANST
201 INTEGER I, J, K, KASE, KC, NZ
202 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
203* ..
204* .. Local Arrays ..
205 INTEGER ISAVE( 3 )
206* ..
207* .. External Subroutines ..
208 EXTERNAL daxpy, dcopy, dlacn2, dtpmv, dtpsv, xerbla
209* ..
210* .. Intrinsic Functions ..
211 INTRINSIC abs, max
212* ..
213* .. External Functions ..
214 LOGICAL LSAME
215 DOUBLE PRECISION DLAMCH
216 EXTERNAL lsame, dlamch
217* ..
218* .. Executable Statements ..
219*
220* Test the input parameters.
221*
222 info = 0
223 upper = lsame( uplo, 'U' )
224 notran = lsame( trans, 'N' )
225 nounit = lsame( diag, 'N' )
226*
227 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
228 info = -1
229 ELSE IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
230 $ lsame( trans, 'C' ) ) THEN
231 info = -2
232 ELSE IF( .NOT.nounit .AND. .NOT.lsame( diag, 'U' ) ) THEN
233 info = -3
234 ELSE IF( n.LT.0 ) THEN
235 info = -4
236 ELSE IF( nrhs.LT.0 ) THEN
237 info = -5
238 ELSE IF( ldb.LT.max( 1, n ) ) THEN
239 info = -8
240 ELSE IF( ldx.LT.max( 1, n ) ) THEN
241 info = -10
242 END IF
243 IF( info.NE.0 ) THEN
244 CALL xerbla( 'DTPRFS', -info )
245 RETURN
246 END IF
247*
248* Quick return if possible
249*
250 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
251 DO 10 j = 1, nrhs
252 ferr( j ) = zero
253 berr( j ) = zero
254 10 CONTINUE
255 RETURN
256 END IF
257*
258 IF( notran ) THEN
259 transt = 'T'
260 ELSE
261 transt = 'N'
262 END IF
263*
264* NZ = maximum number of nonzero elements in each row of A, plus 1
265*
266 nz = n + 1
267 eps = dlamch( 'Epsilon' )
268 safmin = dlamch( 'Safe minimum' )
269 safe1 = nz*safmin
270 safe2 = safe1 / eps
271*
272* Do for each right hand side
273*
274 DO 250 j = 1, nrhs
275*
276* Compute residual R = B - op(A) * X,
277* where op(A) = A or A**T, depending on TRANS.
278*
279 CALL dcopy( n, x( 1, j ), 1, work( n+1 ), 1 )
280 CALL dtpmv( uplo, trans, diag, n, ap, work( n+1 ), 1 )
281 CALL daxpy( n, -one, b( 1, j ), 1, work( n+1 ), 1 )
282*
283* Compute componentwise relative backward error from formula
284*
285* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
286*
287* where abs(Z) is the componentwise absolute value of the matrix
288* or vector Z. If the i-th component of the denominator is less
289* than SAFE2, then SAFE1 is added to the i-th components of the
290* numerator and denominator before dividing.
291*
292 DO 20 i = 1, n
293 work( i ) = abs( b( i, j ) )
294 20 CONTINUE
295*
296 IF( notran ) THEN
297*
298* Compute abs(A)*abs(X) + abs(B).
299*
300 IF( upper ) THEN
301 kc = 1
302 IF( nounit ) THEN
303 DO 40 k = 1, n
304 xk = abs( x( k, j ) )
305 DO 30 i = 1, k
306 work( i ) = work( i ) + abs( ap( kc+i-1 ) )*xk
307 30 CONTINUE
308 kc = kc + k
309 40 CONTINUE
310 ELSE
311 DO 60 k = 1, n
312 xk = abs( x( k, j ) )
313 DO 50 i = 1, k - 1
314 work( i ) = work( i ) + abs( ap( kc+i-1 ) )*xk
315 50 CONTINUE
316 work( k ) = work( k ) + xk
317 kc = kc + k
318 60 CONTINUE
319 END IF
320 ELSE
321 kc = 1
322 IF( nounit ) THEN
323 DO 80 k = 1, n
324 xk = abs( x( k, j ) )
325 DO 70 i = k, n
326 work( i ) = work( i ) + abs( ap( kc+i-k ) )*xk
327 70 CONTINUE
328 kc = kc + n - k + 1
329 80 CONTINUE
330 ELSE
331 DO 100 k = 1, n
332 xk = abs( x( k, j ) )
333 DO 90 i = k + 1, n
334 work( i ) = work( i ) + abs( ap( kc+i-k ) )*xk
335 90 CONTINUE
336 work( k ) = work( k ) + xk
337 kc = kc + n - k + 1
338 100 CONTINUE
339 END IF
340 END IF
341 ELSE
342*
343* Compute abs(A**T)*abs(X) + abs(B).
344*
345 IF( upper ) THEN
346 kc = 1
347 IF( nounit ) THEN
348 DO 120 k = 1, n
349 s = zero
350 DO 110 i = 1, k
351 s = s + abs( ap( kc+i-1 ) )*abs( x( i, j ) )
352 110 CONTINUE
353 work( k ) = work( k ) + s
354 kc = kc + k
355 120 CONTINUE
356 ELSE
357 DO 140 k = 1, n
358 s = abs( x( k, j ) )
359 DO 130 i = 1, k - 1
360 s = s + abs( ap( kc+i-1 ) )*abs( x( i, j ) )
361 130 CONTINUE
362 work( k ) = work( k ) + s
363 kc = kc + k
364 140 CONTINUE
365 END IF
366 ELSE
367 kc = 1
368 IF( nounit ) THEN
369 DO 160 k = 1, n
370 s = zero
371 DO 150 i = k, n
372 s = s + abs( ap( kc+i-k ) )*abs( x( i, j ) )
373 150 CONTINUE
374 work( k ) = work( k ) + s
375 kc = kc + n - k + 1
376 160 CONTINUE
377 ELSE
378 DO 180 k = 1, n
379 s = abs( x( k, j ) )
380 DO 170 i = k + 1, n
381 s = s + abs( ap( kc+i-k ) )*abs( x( i, j ) )
382 170 CONTINUE
383 work( k ) = work( k ) + s
384 kc = kc + n - k + 1
385 180 CONTINUE
386 END IF
387 END IF
388 END IF
389 s = zero
390 DO 190 i = 1, n
391 IF( work( i ).GT.safe2 ) THEN
392 s = max( s, abs( work( n+i ) ) / work( i ) )
393 ELSE
394 s = max( s, ( abs( work( n+i ) )+safe1 ) /
395 $ ( work( i )+safe1 ) )
396 END IF
397 190 CONTINUE
398 berr( j ) = s
399*
400* Bound error from formula
401*
402* norm(X - XTRUE) / norm(X) .le. FERR =
403* norm( abs(inv(op(A)))*
404* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
405*
406* where
407* norm(Z) is the magnitude of the largest component of Z
408* inv(op(A)) is the inverse of op(A)
409* abs(Z) is the componentwise absolute value of the matrix or
410* vector Z
411* NZ is the maximum number of nonzeros in any row of A, plus 1
412* EPS is machine epsilon
413*
414* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
415* is incremented by SAFE1 if the i-th component of
416* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
417*
418* Use DLACN2 to estimate the infinity-norm of the matrix
419* inv(op(A)) * diag(W),
420* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
421*
422 DO 200 i = 1, n
423 IF( work( i ).GT.safe2 ) THEN
424 work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
425 ELSE
426 work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
427 END IF
428 200 CONTINUE
429*
430 kase = 0
431 210 CONTINUE
432 CALL dlacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
433 $ kase, isave )
434 IF( kase.NE.0 ) THEN
435 IF( kase.EQ.1 ) THEN
436*
437* Multiply by diag(W)*inv(op(A)**T).
438*
439 CALL dtpsv( uplo, transt, diag, n, ap, work( n+1 ), 1 )
440 DO 220 i = 1, n
441 work( n+i ) = work( i )*work( n+i )
442 220 CONTINUE
443 ELSE
444*
445* Multiply by inv(op(A))*diag(W).
446*
447 DO 230 i = 1, n
448 work( n+i ) = work( i )*work( n+i )
449 230 CONTINUE
450 CALL dtpsv( uplo, trans, diag, n, ap, work( n+1 ), 1 )
451 END IF
452 GO TO 210
453 END IF
454*
455* Normalize error.
456*
457 lstres = zero
458 DO 240 i = 1, n
459 lstres = max( lstres, abs( x( i, j ) ) )
460 240 CONTINUE
461 IF( lstres.NE.zero )
462 $ ferr( j ) = ferr( j ) / lstres
463*
464 250 CONTINUE
465*
466 RETURN
467*
468* End of DTPRFS
469*
xerbla
subroutine xerbla(srname, info)
Definition cblat2.f:3285
daxpy
subroutine daxpy(n, da, dx, incx, dy, incy)
DAXPY
Definition daxpy.f:89
dcopy
subroutine dcopy(n, dx, incx, dy, incy)
DCOPY
Definition dcopy.f:82
dlacn2
subroutine dlacn2(n, v, x, isgn, est, kase, isave)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition dlacn2.f:136
dlamch
double precision function dlamch(cmach)
DLAMCH
Definition dlamch.f:69
lsame
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
dtpmv
subroutine dtpmv(uplo, trans, diag, n, ap, x, incx)
DTPMV
Definition dtpmv.f:142
dtpsv
subroutine dtpsv(uplo, trans, diag, n, ap, x, incx)
DTPSV
Definition dtpsv.f:144
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