SUBROUTINE ZPBSVX( 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
DOUBLE PRECISION   RCOND
*     ..
*     .. Array Arguments ..
DOUBLE PRECISION   BERR( * ), FERR( * ), RWORK( * ), S( * )
COMPLEX*16         AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
\$                   WORK( * ), X( LDX, * )
*     ..
*
*  Purpose
*  =======
*
*  ZPBSVX 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*16 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*16 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) 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.
*
*  B       (input/output) COMPLEX*16 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*16 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) 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.
*
*  FERR    (output) 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.
*
*  BERR    (output) 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).
*
*  WORK    (workspace) COMPLEX*16 array, dimension (2*N)
*
*  RWORK   (workspace) DOUBLE PRECISION 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 ..
DOUBLE PRECISION   ZERO, ONE
PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
*     ..
*     .. Local Scalars ..
LOGICAL            EQUIL, NOFACT, RCEQU, UPPER
INTEGER            I, INFEQU, J, J1, J2
DOUBLE PRECISION   AMAX, ANORM, BIGNUM, SCOND, SMAX, SMIN, SMLNUM
*     ..
*     .. External Functions ..
LOGICAL            LSAME
DOUBLE PRECISION   DLAMCH, ZLANHB
EXTERNAL           LSAME, DLAMCH, ZLANHB
*     ..
*     .. External Subroutines ..
EXTERNAL           XERBLA, ZCOPY, ZLACPY, ZLAQHB, ZPBCON, ZPBEQU,
\$                   ZPBRFS, ZPBTRF, ZPBTRS
*     ..
*     .. 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 = DLAMCH( '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( 'ZPBSVX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
*        Compute row and column scalings to equilibrate the matrix A.
*
CALL ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
*           Equilibrate the matrix.
*
CALL ZLAQHB( 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 ZCOPY( 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 ZCOPY( J2-J+1, AB( 1, J ), 1, AFB( 1, J ), 1 )
50       CONTINUE
END IF
*
CALL ZPBTRF( 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 = ZLANHB( '1', UPLO, N, KD, AB, LDAB, RWORK )
*
*     Compute the reciprocal of the condition number of A.
*
CALL ZPBCON( UPLO, N, KD, AFB, LDAFB, ANORM, RCOND, WORK, RWORK,
\$             INFO )
*
*     Compute the solution matrix X.
*
CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL ZPBTRS( 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 ZPBRFS( 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.DLAMCH( 'Epsilon' ) )
\$   INFO = N + 1
*
RETURN
*
*     End of ZPBSVX
*
END