LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ cppsvx()

 subroutine cppsvx ( character fact, character uplo, integer n, integer nrhs, complex, dimension( * ) ap, complex, dimension( * ) afp, character equed, real, dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info )

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

Purpose:
``` CPPSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
compute the solution to a complex system of linear equations
A * X = B,
where A is an N-by-N Hermitian 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**H * U ,  if UPLO = 'U', or
A = L * L**H,  if UPLO = 'L',
where U is an upper triangular matrix, L is a lower triangular
matrix, and **H indicates conjugate transpose.

3. If the leading principal minor of order i is not positive,
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 COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian 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 COMPLEX 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**H*U or A = L*L**H, 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**H * U or A = L * L**H 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**H*U or A = L*L**H 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 REAL 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 COMPLEX 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 COMPLEX 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 REAL 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 REAL 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 REAL 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 COMPLEX array, dimension (2*N)` [out] RWORK ` RWORK is REAL 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 principal minor of order i of A is not positive, 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 Hermitian 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 cppsvx.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 REAL RCOND
320* ..
321* .. Array Arguments ..
322 REAL BERR( * ), FERR( * ), RWORK( * ), S( * )
323 COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
324 \$ X( LDX, * )
325* ..
326*
327* =====================================================================
328*
329* .. Parameters ..
330 REAL ZERO, ONE
331 parameter( zero = 0.0e+0, one = 1.0e+0 )
332* ..
333* .. Local Scalars ..
334 LOGICAL EQUIL, NOFACT, RCEQU
335 INTEGER I, INFEQU, J
336 REAL AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
337* ..
338* .. External Functions ..
339 LOGICAL LSAME
340 REAL CLANHP, SLAMCH
341 EXTERNAL lsame, clanhp, slamch
342* ..
343* .. External Subroutines ..
344 EXTERNAL ccopy, clacpy, claqhp, cppcon, cppequ, cpprfs,
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 = slamch( '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( 'CPPSVX', -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 cppequ( uplo, n, ap, s, scond, amax, infequ )
414 IF( infequ.EQ.0 ) THEN
415*
416* Equilibrate the matrix.
417*
418 CALL claqhp( 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**H * U or A = L * L**H.
436*
437 CALL ccopy( n*( n+1 ) / 2, ap, 1, afp, 1 )
438 CALL cpptrf( 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 = clanhp( 'I', uplo, n, ap, rwork )
451*
452* Compute the reciprocal of the condition number of A.
453*
454 CALL cppcon( uplo, n, afp, anorm, rcond, work, rwork, info )
455*
456* Compute the solution matrix X.
457*
458 CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
459 CALL cpptrs( 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 cpprfs( uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr,
465 \$ work, rwork, 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.slamch( 'Epsilon' ) )
484 \$ info = n + 1
485*
486 RETURN
487*
488* End of CPPSVX
489*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine ccopy(n, cx, incx, cy, incy)
CCOPY
Definition ccopy.f:81
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
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
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 claqhp(uplo, n, ap, s, scond, amax, equed)
CLAQHP scales a Hermitian matrix stored in packed form.
Definition claqhp.f:126
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine cppcon(uplo, n, ap, anorm, rcond, work, rwork, info)
CPPCON
Definition cppcon.f:118
subroutine cppequ(uplo, n, ap, s, scond, amax, info)
CPPEQU
Definition cppequ.f:117
subroutine cpprfs(uplo, n, nrhs, ap, afp, b, ldb, x, ldx, ferr, berr, work, rwork, info)
CPPRFS
Definition cpprfs.f:171
subroutine cpptrf(uplo, n, ap, info)
CPPTRF
Definition cpptrf.f:119
subroutine cpptrs(uplo, n, nrhs, ap, b, ldb, info)
CPPTRS
Definition cpptrs.f:108
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