Move LAPACK trunk into position.

1 files changed, 421 insertions, 0 deletions

diff --git a/SRC/cpbsvx.f b/SRC/cpbsvx.f
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+      SUBROUTINE CPBSVX( FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
+     $                   EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
+     $                   WORK, RWORK, INFO )
+*
+*  -- LAPACK driver routine (version 3.1) --
+*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
+*     November 2006
+*
+*     .. Scalar Arguments ..
+      CHARACTER          EQUED, FACT, UPLO
+      INTEGER            INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
+      REAL               RCOND
+*     ..
+*     .. Array Arguments ..
+      REAL               BERR( * ), FERR( * ), RWORK( * ), S( * )
+      COMPLEX            AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
+     $                   WORK( * ), X( LDX, * )
+*     ..
+*
+*  Purpose
+*  =======
+*
+*  CPBSVX 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 band matrix 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 band matrix, and L is a lower
+*     triangular band 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.
+*
+*  Arguments
+*  =========
+*
+*  FACT    (input) 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, AFB contains the factored form of A.
+*                  If EQUED = 'Y', the matrix A has been equilibrated
+*                  with scaling factors given by S.  AB and AFB will not
+*                  be modified.
+*          = 'N':  The matrix A will be copied to AFB and factored.
+*          = 'E':  The matrix A will be equilibrated if necessary, then
+*                  copied to AFB and factored.
+*
+*  UPLO    (input) CHARACTER*1
+*          = 'U':  Upper triangle of A is stored;
+*          = 'L':  Lower triangle of A is stored.
+*
+*  N       (input) INTEGER
+*          The number of linear equations, i.e., the order of the
+*          matrix A.  N >= 0.
+*
+*  KD      (input) INTEGER
+*          The number of superdiagonals of the matrix A if UPLO = 'U',
+*          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
+*
+*  NRHS    (input) INTEGER
+*          The number of right-hand sides, i.e., the number of columns
+*          of the matrices B and X.  NRHS >= 0.
+*
+*  AB      (input/output) COMPLEX array, dimension (LDAB,N)
+*          On entry, the upper or lower triangle of the Hermitian band
+*          matrix A, stored in the first KD+1 rows of the 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 j-th column of the array AB as follows:
+*          if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
+*          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
+*          See below for further details.
+*
+*          On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
+*          diag(S)*A*diag(S).
+*
+*  LDAB    (input) INTEGER
+*          The leading dimension of the array A.  LDAB >= KD+1.
+*
+*  AFB     (input or output) COMPLEX array, dimension (LDAFB,N)
+*          If FACT = 'F', then AFB 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 of the band matrix
+*          A, in the same storage format as A (see AB).  If EQUED = 'Y',
+*          then AFB is the factored form of the equilibrated matrix A.
+*
+*          If FACT = 'N', then AFB 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.
+*
+*          If FACT = 'E', then AFB 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 A for the form of the
+*          equilibrated matrix).
+*
+*  LDAFB   (input) INTEGER
+*          The leading dimension of the array AFB.  LDAFB >= KD+1.
+*
+*  EQUED   (input or output) 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.
+*
+*  S       (input or output) 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.
+*
+*  B       (input/output) 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.
+*
+*  LDB     (input) INTEGER
+*          The leading dimension of the array B.  LDB >= max(1,N).
+*
+*  X       (output) 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.
+*
+*  LDX     (input) INTEGER
+*          The leading dimension of the array X.  LDX >= max(1,N).
+*
+*  RCOND   (output) 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.
+*
+*  FERR    (output) 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.
+*
+*  BERR    (output) 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).
+*
+*  WORK    (workspace) COMPLEX array, dimension (2*N)
+*
+*  RWORK   (workspace) REAL array, dimension (N)
+*
+*  INFO    (output) 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 band storage scheme is illustrated by the following example, when
+*  N = 6, KD = 2, and UPLO = 'U':
+*
+*  Two-dimensional storage of the Hermitian matrix A:
+*
+*     a11  a12  a13
+*          a22  a23  a24
+*               a33  a34  a35
+*                    a44  a45  a46
+*                         a55  a56
+*     (aij=conjg(aji))         a66
+*
+*  Band storage of the upper triangle of A:
+*
+*      *    *   a13  a24  a35  a46
+*      *   a12  a23  a34  a45  a56
+*     a11  a22  a33  a44  a55  a66
+*
+*  Similarly, if UPLO = 'L' the format of A is as follows:
+*
+*     a11  a22  a33  a44  a55  a66
+*     a21  a32  a43  a54  a65   *
+*     a31  a42  a53  a64   *    *
+*
+*  Array elements marked * are not used by the routine.
+*
+*  =====================================================================
+*
+*     .. Parameters ..
+      REAL               ZERO, ONE
+      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
+*     ..
+*     .. Local Scalars ..
+      LOGICAL            EQUIL, NOFACT, RCEQU, UPPER
+      INTEGER            I, INFEQU, J, J1, J2
+      REAL               AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
+*     ..
+*     .. External Functions ..
+      LOGICAL            LSAME
+      REAL               CLANHB, SLAMCH
+      EXTERNAL           LSAME, CLANHB, SLAMCH
+*     ..
+*     .. External Subroutines ..
+      EXTERNAL           CCOPY, CLACPY, CLAQHB, CPBCON, CPBEQU, CPBRFS,
+     $                   CPBTRF, CPBTRS, XERBLA
+*     ..
+*     .. Intrinsic Functions ..
+      INTRINSIC          MAX, MIN
+*     ..
+*     .. Executable Statements ..
+*
+      INFO = 0
+      NOFACT = LSAME( FACT, 'N' )
+      EQUIL = LSAME( FACT, 'E' )
+      UPPER = LSAME( UPLO, 'U' )
+      IF( NOFACT .OR. EQUIL ) THEN
+         EQUED = 'N'
+         RCEQU = .FALSE.
+      ELSE
+         RCEQU = LSAME( EQUED, 'Y' )
+         SMLNUM = SLAMCH( 'Safe minimum' )
+         BIGNUM = ONE / SMLNUM
+      END IF
+*
+*     Test the input parameters.
+*
+      IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
+     $     THEN
+         INFO = -1
+      ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
+         INFO = -2
+      ELSE IF( N.LT.0 ) THEN
+         INFO = -3
+      ELSE IF( KD.LT.0 ) THEN
+         INFO = -4
+      ELSE IF( NRHS.LT.0 ) THEN
+         INFO = -5
+      ELSE IF( LDAB.LT.KD+1 ) THEN
+         INFO = -7
+      ELSE IF( LDAFB.LT.KD+1 ) THEN
+         INFO = -9
+      ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
+     $         ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
+         INFO = -10
+      ELSE
+         IF( RCEQU ) THEN
+            SMIN = BIGNUM
+            SMAX = ZERO
+            DO 10 J = 1, N
+               SMIN = MIN( SMIN, S( J ) )
+               SMAX = MAX( SMAX, S( J ) )
+   10       CONTINUE
+            IF( SMIN.LE.ZERO ) THEN
+               INFO = -11
+            ELSE IF( N.GT.0 ) THEN
+               SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
+            ELSE
+               SCOND = ONE
+            END IF
+         END IF
+         IF( INFO.EQ.0 ) THEN
+            IF( LDB.LT.MAX( 1, N ) ) THEN
+               INFO = -13
+            ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
+               INFO = -15
+            END IF
+         END IF
+      END IF
+*
+      IF( INFO.NE.0 ) THEN
+         CALL XERBLA( 'CPBSVX', -INFO )
+         RETURN
+      END IF
+*
+      IF( EQUIL ) THEN
+*
+*        Compute row and column scalings to equilibrate the matrix A.
+*
+         CALL CPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
+         IF( INFEQU.EQ.0 ) THEN
+*
+*           Equilibrate the matrix.
+*
+            CALL CLAQHB( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED )
+            RCEQU = LSAME( EQUED, 'Y' )
+         END IF
+      END IF
+*
+*     Scale the right-hand side.
+*
+      IF( RCEQU ) THEN
+         DO 30 J = 1, NRHS
+            DO 20 I = 1, N
+               B( I, J ) = S( I )*B( I, J )
+   20       CONTINUE
+   30    CONTINUE
+      END IF
+*
+      IF( NOFACT .OR. EQUIL ) THEN
+*
+*        Compute the Cholesky factorization A = U'*U or A = L*L'.
+*
+         IF( UPPER ) THEN
+            DO 40 J = 1, N
+               J1 = MAX( J-KD, 1 )
+               CALL CCOPY( J-J1+1, AB( KD+1-J+J1, J ), 1,
+     $                     AFB( KD+1-J+J1, J ), 1 )
+   40       CONTINUE
+         ELSE
+            DO 50 J = 1, N
+               J2 = MIN( J+KD, N )
+               CALL CCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
+   50       CONTINUE
+         END IF
+*
+         CALL CPBTRF( UPLO, N, KD, AFB, LDAFB, INFO )
+*
+*        Return if INFO is non-zero.
+*
+         IF( INFO.GT.0 )THEN
+            RCOND = ZERO
+            RETURN
+         END IF
+      END IF
+*
+*     Compute the norm of the matrix A.
+*
+      ANORM = CLANHB( '1', UPLO, N, KD, AB, LDAB, RWORK )
+*
+*     Compute the reciprocal of the condition number of A.
+*
+      CALL CPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, RWORK,
+     $             INFO )
+*
+*     Compute the solution matrix X.
+*
+      CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
+      CALL CPBTRS( UPLO, N, KD, NRHS, AFB, LDAFB, X, LDX, INFO )
+*
+*     Use iterative refinement to improve the computed solution and
+*     compute error bounds and backward error estimates for it.
+*
+      CALL CPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B, LDB, X,
+     $             LDX, FERR, BERR, WORK, RWORK, INFO )
+*
+*     Transform the solution matrix X to a solution of the original
+*     system.
+*
+      IF( RCEQU ) THEN
+         DO 70 J = 1, NRHS
+            DO 60 I = 1, N
+               X( I, J ) = S( I )*X( I, J )
+   60       CONTINUE
+   70    CONTINUE
+         DO 80 J = 1, NRHS
+            FERR( J ) = FERR( J ) / SCOND
+   80    CONTINUE
+      END IF
+*
+*     Set INFO = N+1 if the matrix is singular to working precision.
+*
+      IF( RCOND.LT.SLAMCH( 'Epsilon' ) )
+     $   INFO = N + 1
+*
+      RETURN
+*
+*     End of CPBSVX
+*
+      END