 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ dppsvx()

 subroutine dppsvx ( character FACT, character UPLO, integer N, integer NRHS, double precision, dimension( * ) AP, double precision, dimension( * ) AFP, character EQUED, double precision, dimension( * ) S, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldx, * ) X, integer LDX, double precision RCOND, double precision, dimension( * ) FERR, double precision, dimension( * ) BERR, double precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

DPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:
``` DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to
compute the solution to a real system of linear equations
A * X = B,
where A is an N-by-N symmetric positive definite matrix stored in
packed format and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also
provided.```
Description:
``` The following steps are performed:

1. If FACT = 'E', real scaling factors are computed to equilibrate
the system:
diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B
Whether or not the system will be equilibrated depends on the
scaling of the matrix A, but if equilibration is used, A is
overwritten by diag(S)*A*diag(S) and B by diag(S)*B.

2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
factor the matrix A (after equilibration if FACT = 'E') as
A = U**T* U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',
where U is an upper triangular matrix and L is a lower triangular
matrix.

3. If the leading i-by-i principal minor is not positive definite,
then the routine returns with INFO = i. Otherwise, the factored
form of A is used to estimate the condition number of the matrix
A.  If the reciprocal of the condition number is less than machine
precision, INFO = N+1 is returned as a warning, but the routine
still goes on to solve for X and compute error bounds as
described below.

4. The system of equations is solved for X using the factored form
of A.

5. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.

6. If equilibration was used, the matrix X is premultiplied by
diag(S) so that it solves the original system before
equilibration.```
Parameters
 [in] FACT ``` FACT is CHARACTER*1 Specifies whether or not the factored form of the matrix A is supplied on entry, and if not, whether the matrix A should be equilibrated before it is factored. = 'F': On entry, AFP contains the factored form of A. If EQUED = 'Y', the matrix A has been equilibrated with scaling factors given by S. AP and AFP will not be modified. = 'N': The matrix A will be copied to AFP and factored. = 'E': The matrix A will be equilibrated if necessary, then copied to AFP and factored.``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The number of linear equations, i.e., 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,out] AP ``` AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array, except if FACT = 'F' and EQUED = 'Y', then A must contain the equilibrated matrix diag(S)*A*diag(S). 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)*(2n-j)/2) = A(i,j) for j<=i<=n. See below for further details. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by diag(S)*A*diag(S).``` [in,out] AFP ``` AFP is DOUBLE PRECISION array, dimension (N*(N+1)/2) If FACT = 'F', then AFP is an input argument and on entry contains the triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, in the same storage format as A. If EQUED .ne. 'N', then AFP is the factored form of the equilibrated matrix A. If FACT = 'N', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T * U or A = L * L**T of the original matrix A. If FACT = 'E', then AFP is an output argument and on exit returns the triangular factor U or L from the Cholesky factorization A = U**T * U or A = L * L**T of the equilibrated matrix A (see the description of AP for the form of the equilibrated matrix).``` [in,out] EQUED ``` EQUED is CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'Y': Equilibration was done, i.e., A has been replaced by diag(S) * A * diag(S). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument.``` [in,out] S ``` S is DOUBLE PRECISION array, dimension (N) The scale factors for A; not accessed if EQUED = 'N'. S is an input argument if FACT = 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED = 'Y', each element of S must be positive.``` [in,out] B ``` B is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the N-by-NRHS right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', B is overwritten by diag(S) * B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [out] X ``` X is DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to the original system of equations. Note that if EQUED = 'Y', A and B are modified on exit, and the solution to the equilibrated system is inv(diag(S))*X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] RCOND ``` RCOND is DOUBLE PRECISION The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0.``` [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 > 0: if INFO = i, and i is <= N: the leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed. RCOND = 0 is returned. = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest.```
Further Details:
```  The packed storage scheme is illustrated by the following example
when N = 4, UPLO = 'U':

Two-dimensional storage of the symmetric matrix A:

a11 a12 a13 a14
a22 a23 a24
a33 a34     (aij = conjg(aji))
a44

Packed storage of the upper triangle of A:

AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]```

Definition at line 309 of file dppsvx.f.

311 *
312 * -- LAPACK driver routine --
313 * -- LAPACK is a software package provided by Univ. of Tennessee, --
314 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
315 *
316 * .. Scalar Arguments ..
317  CHARACTER EQUED, FACT, UPLO
318  INTEGER INFO, LDB, LDX, N, NRHS
319  DOUBLE PRECISION RCOND
320 * ..
321 * .. Array Arguments ..
322  INTEGER IWORK( * )
323  DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
324  \$ FERR( * ), S( * ), WORK( * ), X( LDX, * )
325 * ..
326 *
327 * =====================================================================
328 *
329 * .. Parameters ..
330  DOUBLE PRECISION ZERO, ONE
331  parameter( zero = 0.0d+0, one = 1.0d+0 )
332 * ..
333 * .. Local Scalars ..
334  LOGICAL EQUIL, NOFACT, RCEQU
335  INTEGER I, INFEQU, J
336  DOUBLE PRECISION AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
337 * ..
338 * .. External Functions ..
339  LOGICAL LSAME
340  DOUBLE PRECISION DLAMCH, DLANSP
341  EXTERNAL lsame, dlamch, dlansp
342 * ..
343 * .. External Subroutines ..
344  EXTERNAL dcopy, dlacpy, dlaqsp, dppcon, dppequ, dpprfs,
345  \$ dpptrf, dpptrs, xerbla
346 * ..
347 * .. Intrinsic Functions ..
348  INTRINSIC max, min
349 * ..
350 * .. Executable Statements ..
351 *
352  info = 0
353  nofact = lsame( fact, 'N' )
354  equil = lsame( fact, 'E' )
355  IF( nofact .OR. equil ) THEN
356  equed = 'N'
357  rcequ = .false.
358  ELSE
359  rcequ = lsame( equed, 'Y' )
360  smlnum = dlamch( 'Safe minimum' )
361  bignum = one / smlnum
362  END IF
363 *
364 * Test the input parameters.
365 *
366  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.lsame( fact, 'F' ) )
367  \$ THEN
368  info = -1
369  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
370  \$ THEN
371  info = -2
372  ELSE IF( n.LT.0 ) THEN
373  info = -3
374  ELSE IF( nrhs.LT.0 ) THEN
375  info = -4
376  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
377  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
378  info = -7
379  ELSE
380  IF( rcequ ) THEN
381  smin = bignum
382  smax = zero
383  DO 10 j = 1, n
384  smin = min( smin, s( j ) )
385  smax = max( smax, s( j ) )
386  10 CONTINUE
387  IF( smin.LE.zero ) THEN
388  info = -8
389  ELSE IF( n.GT.0 ) THEN
390  scond = max( smin, smlnum ) / min( smax, bignum )
391  ELSE
392  scond = one
393  END IF
394  END IF
395  IF( info.EQ.0 ) THEN
396  IF( ldb.LT.max( 1, n ) ) THEN
397  info = -10
398  ELSE IF( ldx.LT.max( 1, n ) ) THEN
399  info = -12
400  END IF
401  END IF
402  END IF
403 *
404  IF( info.NE.0 ) THEN
405  CALL xerbla( 'DPPSVX', -info )
406  RETURN
407  END IF
408 *
409  IF( equil ) THEN
410 *
411 * Compute row and column scalings to equilibrate the matrix A.
412 *
413  CALL dppequ( uplo, n, ap, s, scond, amax, infequ )
414  IF( infequ.EQ.0 ) THEN
415 *
416 * Equilibrate the matrix.
417 *
418  CALL dlaqsp( uplo, n, ap, s, scond, amax, equed )
419  rcequ = lsame( equed, 'Y' )
420  END IF
421  END IF
422 *
423 * Scale the right-hand side.
424 *
425  IF( rcequ ) THEN
426  DO 30 j = 1, nrhs
427  DO 20 i = 1, n
428  b( i, j ) = s( i )*b( i, j )
429  20 CONTINUE
430  30 CONTINUE
431  END IF
432 *
433  IF( nofact .OR. equil ) THEN
434 *
435 * Compute the Cholesky factorization A = U**T * U or A = L * L**T.
436 *
437  CALL dcopy( n*( n+1 ) / 2, ap, 1, afp, 1 )
438  CALL dpptrf( uplo, n, afp, info )
439 *
440 * Return if INFO is non-zero.
441 *
442  IF( info.GT.0 )THEN
443  rcond = zero
444  RETURN
445  END IF
446  END IF
447 *
448 * Compute the norm of the matrix A.
449 *
450  anorm = dlansp( 'I', uplo, n, ap, work )
451 *
452 * Compute the reciprocal of the condition number of A.
453 *
454  CALL dppcon( uplo, n, afp, anorm, rcond, work, iwork, info )
455 *
456 * Compute the solution matrix X.
457 *
458  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
459  CALL dpptrs( uplo, n, nrhs, afp, x, ldx, info )
460 *
461 * Use iterative refinement to improve the computed solution and
462 * compute error bounds and backward error estimates for it.
463 *
464  CALL dpprfs( uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr,
465  \$ work, iwork, info )
466 *
467 * Transform the solution matrix X to a solution of the original
468 * system.
469 *
470  IF( rcequ ) THEN
471  DO 50 j = 1, nrhs
472  DO 40 i = 1, n
473  x( i, j ) = s( i )*x( i, j )
474  40 CONTINUE
475  50 CONTINUE
476  DO 60 j = 1, nrhs
477  ferr( j ) = ferr( j ) / scond
478  60 CONTINUE
479  END IF
480 *
481 * Set INFO = N+1 if the matrix is singular to working precision.
482 *
483  IF( rcond.LT.dlamch( 'Epsilon' ) )
484  \$ info = n + 1
485 *
486  RETURN
487 *
488 * End of DPPSVX
489 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dlaqsp(UPLO, N, AP, S, SCOND, AMAX, EQUED)
DLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppeq...
Definition: dlaqsp.f:125
double precision function dlansp(NORM, UPLO, N, AP, WORK)
DLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansp.f:114
subroutine dpptrf(UPLO, N, AP, INFO)
DPPTRF
Definition: dpptrf.f:119
subroutine dppequ(UPLO, N, AP, S, SCOND, AMAX, INFO)
DPPEQU
Definition: dppequ.f:116
subroutine dpprfs(UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DPPRFS
Definition: dpprfs.f:171
subroutine dppcon(UPLO, N, AP, ANORM, RCOND, WORK, IWORK, INFO)
DPPCON
Definition: dppcon.f:118
subroutine dpptrs(UPLO, N, NRHS, AP, B, LDB, INFO)
DPPTRS
Definition: dpptrs.f:108
Here is the call graph for this function:
Here is the caller graph for this function: