Purpose
=======
LA_PBSVX computes the solution to a linear system of equations
A*X = B, where A has band form and is real symmetric or complex
Hermitian and, in either case, positive definite, and where X and B
are rectangular matrices or vectors.
LA_PBSVX can also optionally equilibrate the system if A is poorly
scaled, estimate the condition number of (the equilibrated) A, and
compute error bounds.
=========
SUBROUTINE LA_PBSVX( AB, B, X, UPLO=uplo, AFB=afb, FACT=fact, &
EQUED=equed, S=s, FERR=ferr, BERR=berr, &
RCOND=rcond, INFO=info )
(), INTENT(INOUT) :: AB(:,:),
(), INTENT(OUT) ::
CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: UPLO
(), INTENT(INOUT), OPTIONAL :: AFB(:,:)
CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: FACT
CHARACTER(LEN=1), INTENT(INOUT), OPTIONAL :: EQUED
REAL(), INTENT(INOUT), OPTIONAL :: S(:)
REAL(), INTENT(OUT), OPTIONAL ::
REAL(), INTENT(OUT), OPTIONAL :: RCOND
INTEGER, INTENT(OUT), OPTIONAL :: INFO
where
::= REAL | COMPLEX
::= KIND(1.0) | KIND(1.0D0)
::= B(:,:) | B(:)
::= X(:,:) | X(:)
::= FERR(:), BERR(:) | FERR, BERR
Arguments
=========
AB (input/output) REAL or COMPLEX array, shape (:,:) with
size(AB,1) = kd + 1 and size(AB,2) = n, where kd is the number
of superdiagonals or subdiagonals and n is the order of A.
On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
triangle of matrix A, or its equilibration, in band storage.
The (kd + 1) diagonals of A are stored in the rows of AB so
that the j-th column of A is stored in the j-th column of AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j
1<=j<=n
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd)
1<=j<=n.
On exit, if FACT = 'E', then the equilibrated version of A is
stored in AB; otherwise, AB is unchanged.
B (input/output) REAL or COMPLEX array, shape (:,:) with
size(B,1) = n or shape (:) with size(B) = n.
On entry, the matrix B.
On exit, the scaled version of B if the system has been
equilibrated; otherwise, B is unchanged.
X (output) REAL or COMPLEX array, shape (:,:) with size(X,1)=n
and size(X,2) = size(B,2), or shape (:) with size(X) = n.
The solution matrix X .
UPLO Optional (input) CHARACTER(LEN=1).
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
Default value: 'U'.
AFB Optional (input or output) REAL or COMPLEX array, shape (:)
with the same size as AB.
If FACT = 'F' then AFB is an input argument that contains the
factor U or L from the Cholesky factorization of (the
equilibrated) A, in the same storage format as A, returned by
a previous call to LA_PBSVX.
If FACT /= 'F' then AFB is an output argument that contains
the factor U or L from the Cholesky factorization of (the
equilibrated) A in the same storage format as A.
FACT Optional (input) CHARACTER(LEN=1).
Specifies whether the factored form of the matrix A is supplied
on entry, and if not, whether A should be equilibrated before
it is factored.
= 'N': The matrix A will be copied to AFB and factored (no
equilibration).
= 'E': The matrix A will be equilibrated, then copied to
AFB and factored.
= 'F': AFB contains the factored form of (the equilibrated)
A.
Default value: 'N'.
EQUED Optional (input or output) CHARACTER(LEN=1).
Specifies the form of equilibration that was done.
EQUED is an input argument if FACT = 'F', otherwise it is an
output argument
= 'N': No equilibration (always true if FACT = 'N').
= 'Y': Equilibration, i.e., A has been premultiplied and
postmultiplied by diag(S).
Default value: 'N'.
S Optional (input or output) REAL array, shape (:) with size(S)
= size(A,1).
The scaling factors for A.
S is an input argument if FACT = 'F' and EQUED = 'Y'.
S is an output argument if FACT = 'E' and EQUED = 'Y'.
FERR Optional (output) REAL array of shape (:), with size(FERR) =
size(X,2), or REAL scalar.
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 Optional (output) REAL array of shape (:), with size(BERR) =
size(X,2), or REAL scalar.
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).
RCOND Optional (output) REAL
The estimate of the reciprocal condition number of (the
equilibrated) A. If RCOND is less than the machine precision,
the matrix is singular to working precision. This condition is
indicated by a return code of INFO > 0.
INFO Optional (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 (the equilibrated) A
is not positive definite, so the factorization could
not be completed and the solution and error bounds
could not be computed. RCOND= 0 is returned.
= n+1: U or L is nonsingular, but RCOND is less than
machine precision, so the matrix is singular to
working precision. Nevertheless, the solution and
error bounds are computed because the computed solution
can be more accurate than the value of RCOND would
suggest.
If INFO is not present and an error occurs, then the program
is terminated with an error message.