.TH CPBSVX l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
CPBSVX - use 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,
.SH SYNOPSIS
.TP 19
SUBROUTINE CPBSVX(
FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB,
EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR,
WORK, RWORK, INFO )
.TP 19
.ti +4
CHARACTER
EQUED, FACT, UPLO
.TP 19
.ti +4
INTEGER
INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
.TP 19
.ti +4
REAL
RCOND
.TP 19
.ti +4
REAL
BERR( * ), FERR( * ), RWORK( * ), S( * )
.TP 19
.ti +4
COMPLEX
AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
WORK( * ), X( LDX, * )
.SH 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.
.br

Error bounds on the solution and a condition estimate are also
provided.
.br

.SH DESCRIPTION
The following steps are performed:
.br

1. If FACT = 'E', real scaling factors are computed to equilibrate
   the system:
.br
      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.
.br

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
.br
      A = L * L**H,  if UPLO = 'L',
.br
   where U is an upper triangular band matrix, and L is a lower
   triangular band matrix.
.br

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.
.br

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

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

6. If equilibration was used, the matrix X is premultiplied by
   diag(S) so that it solves the original system before
.br
   equilibration.
.br

.SH ARGUMENTS
.TP 8
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.
.br
= 'E':  The matrix A will be equilibrated if necessary, then
copied to AFB and factored.
.TP 8
UPLO    (input) CHARACTER*1
= 'U':  Upper triangle of A is stored;
.br
= 'L':  Lower triangle of A is stored.
.TP 8
N       (input) INTEGER
The number of linear equations, i.e., the order of the
matrix A.  N >= 0.
.TP 8
KD      (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
.TP 8
NRHS    (input) INTEGER
The number of right-hand sides, i.e., the number of columns
of the matrices B and X.  NRHS >= 0.
.TP 8
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).
.TP 8
LDAB    (input) INTEGER
The leading dimension of the array A.  LDAB >= KD+1.
.TP 8
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).
.TP 8
LDAFB   (input) INTEGER
The leading dimension of the array AFB.  LDAFB >= KD+1.
.TP 8
EQUED   (input or output) CHARACTER*1
Specifies the form of equilibration that was done.
= 'N':  No equilibration (always true if FACT = 'N').
.br
= '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.
.TP 8
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.
.TP 8
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.
.TP 8
LDB     (input) INTEGER
The leading dimension of the array B.  LDB >= max(1,N).
.TP 8
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.
.TP 8
LDX     (input) INTEGER
The leading dimension of the array X.  LDX >= max(1,N).
.TP 8
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.
.TP 8
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.
.TP 8
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).
.TP 8
WORK    (workspace) COMPLEX array, dimension (2*N)
.TP 8
RWORK   (workspace) REAL array, dimension (N)
.TP 8
INFO    (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value
.br
> 0: if INFO = i, and i is
.br
<= 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.
.SH FURTHER DETAILS
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = 'U':
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Two-dimensional storage of the Hermitian matrix A:
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   a11  a12  a13
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        a22  a23  a24
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             a33  a34  a35
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                  a44  a45  a46
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                       a55  a56
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   (aij=conjg(aji))         a66
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Band storage of the upper triangle of A:
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    *    *   a13  a24  a35  a46
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    *   a12  a23  a34  a45  a56
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   a11  a22  a33  a44  a55  a66
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Similarly, if UPLO = 'L' the format of A is as follows:
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   a11  a22  a33  a44  a55  a66
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   a21  a32  a43  a54  a65   *
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   a31  a42  a53  a64   *    *
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Array elements marked * are not used by the routine.
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