LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ chpsvx()

subroutine chpsvx ( character  FACT,
character  UPLO,
integer  N,
integer  NRHS,
complex, dimension( * )  AP,
complex, dimension( * )  AFP,
integer, dimension( * )  IPIV,
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 
)

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

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

Purpose:
 CHPSVX uses the diagonal pivoting factorization A = U*D*U**H or
 A = L*D*L**H to compute the solution to a complex system of linear
 equations A * X = B, where A is an N-by-N Hermitian 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 = 'N', the diagonal pivoting method is used to factor A as
       A = U * D * U**H,  if UPLO = 'U', or
       A = L * D * L**H,  if UPLO = 'L',
    where U (or L) is a product of permutation and unit upper (lower)
    triangular matrices and D is Hermitian and block diagonal with
    1-by-1 and 2-by-2 diagonal blocks.

 2. If some D(i,i)=0, so that D is exactly singular, 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.

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

 4. Iterative refinement is applied to improve the computed solution
    matrix and calculate error bounds and backward error estimates
    for it.
Parameters
[in]FACT
          FACT is CHARACTER*1
          Specifies whether or not the factored form of A has been
          supplied on entry.
          = 'F':  On entry, AFP and IPIV contain the factored form of
                  A.  AFP and IPIV will not be modified.
          = 'N':  The matrix A will be 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]AP
          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangle of the Hermitian 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.
          See below for further details.
[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 block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as
          a packed triangular matrix in the same storage format as A.

          If FACT = 'N', then AFP is an output argument and on exit
          contains the block diagonal matrix D and the multipliers used
          to obtain the factor U or L from the factorization
          A = U*D*U**H or A = L*D*L**H as computed by CHPTRF, stored as
          a packed triangular matrix in the same storage format as A.
[in,out]IPIV
          IPIV is INTEGER array, dimension (N)
          If FACT = 'F', then IPIV is an input argument and on entry
          contains details of the interchanges and the block structure
          of D, as determined by CHPTRF.
          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
          interchanged and D(k,k) is a 1-by-1 diagonal block.
          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

          If FACT = 'N', then IPIV is an output argument and on exit
          contains details of the interchanges and the block structure
          of D, as determined by CHPTRF.
[in]B
          B is COMPLEX array, dimension (LDB,NRHS)
          The N-by-NRHS right hand side matrix 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.
[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.  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:  D(i,i) is exactly zero.  The factorization
                       has been completed but the factor D is exactly
                       singular, so the solution and error bounds could
                       not be computed. RCOND = 0 is returned.
                = N+1: D 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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 275 of file chpsvx.f.

277 *
278 * -- LAPACK driver routine --
279 * -- LAPACK is a software package provided by Univ. of Tennessee, --
280 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
281 *
282 * .. Scalar Arguments ..
283  CHARACTER FACT, UPLO
284  INTEGER INFO, LDB, LDX, N, NRHS
285  REAL RCOND
286 * ..
287 * .. Array Arguments ..
288  INTEGER IPIV( * )
289  REAL BERR( * ), FERR( * ), RWORK( * )
290  COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
291  $ X( LDX, * )
292 * ..
293 *
294 * =====================================================================
295 *
296 * .. Parameters ..
297  REAL ZERO
298  parameter( zero = 0.0e+0 )
299 * ..
300 * .. Local Scalars ..
301  LOGICAL NOFACT
302  REAL ANORM
303 * ..
304 * .. External Functions ..
305  LOGICAL LSAME
306  REAL CLANHP, SLAMCH
307  EXTERNAL lsame, clanhp, slamch
308 * ..
309 * .. External Subroutines ..
310  EXTERNAL ccopy, chpcon, chprfs, chptrf, chptrs, clacpy,
311  $ xerbla
312 * ..
313 * .. Intrinsic Functions ..
314  INTRINSIC max
315 * ..
316 * .. Executable Statements ..
317 *
318 * Test the input parameters.
319 *
320  info = 0
321  nofact = lsame( fact, 'N' )
322  IF( .NOT.nofact .AND. .NOT.lsame( fact, 'F' ) ) THEN
323  info = -1
324  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) )
325  $ THEN
326  info = -2
327  ELSE IF( n.LT.0 ) THEN
328  info = -3
329  ELSE IF( nrhs.LT.0 ) THEN
330  info = -4
331  ELSE IF( ldb.LT.max( 1, n ) ) THEN
332  info = -9
333  ELSE IF( ldx.LT.max( 1, n ) ) THEN
334  info = -11
335  END IF
336  IF( info.NE.0 ) THEN
337  CALL xerbla( 'CHPSVX', -info )
338  RETURN
339  END IF
340 *
341  IF( nofact ) THEN
342 *
343 * Compute the factorization A = U*D*U**H or A = L*D*L**H.
344 *
345  CALL ccopy( n*( n+1 ) / 2, ap, 1, afp, 1 )
346  CALL chptrf( uplo, n, afp, ipiv, info )
347 *
348 * Return if INFO is non-zero.
349 *
350  IF( info.GT.0 )THEN
351  rcond = zero
352  RETURN
353  END IF
354  END IF
355 *
356 * Compute the norm of the matrix A.
357 *
358  anorm = clanhp( 'I', uplo, n, ap, rwork )
359 *
360 * Compute the reciprocal of the condition number of A.
361 *
362  CALL chpcon( uplo, n, afp, ipiv, anorm, rcond, work, info )
363 *
364 * Compute the solution vectors X.
365 *
366  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
367  CALL chptrs( uplo, n, nrhs, afp, ipiv, x, ldx, info )
368 *
369 * Use iterative refinement to improve the computed solutions and
370 * compute error bounds and backward error estimates for them.
371 *
372  CALL chprfs( uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr,
373  $ berr, work, rwork, info )
374 *
375 * Set INFO = N+1 if the matrix is singular to working precision.
376 *
377  IF( rcond.LT.slamch( 'Epsilon' ) )
378  $ info = n + 1
379 *
380  RETURN
381 *
382 * End of CHPSVX
383 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
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 chptrs(UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
CHPTRS
Definition: chptrs.f:115
subroutine chptrf(UPLO, N, AP, IPIV, INFO)
CHPTRF
Definition: chptrf.f:159
subroutine chpcon(UPLO, N, AP, IPIV, ANORM, RCOND, WORK, INFO)
CHPCON
Definition: chpcon.f:118
subroutine chprfs(UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CHPRFS
Definition: chprfs.f:180
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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