Purpose
=======
LA_GBSVX computes the solution to a real or complex linear system of
equations of the form A*X = B, A^T*X = B or A^H*X = B, where A is a
square band matrix and X and B are rectangular matrices or vectors.
LA_GBSVX can also optionally equilibrate the system if A is poorly
scaled, estimate the condition number of (the equilibrated) A, return
the pivot growth factor, and compute error bounds.
=========
SUBROUTINE LA_GBSVX( AB, B, X, KL=kl, AFB=afb, IPIV=ipiv, &
FACT=fact, TRANS=trans, EQUED=equed, R=r, C=c, &
FERR=ferr, BERR=berr, RCOND=rcond, &
RPVGRW=rpvgrw, INFO=info )
(), INTENT(INOUT) :: AB(:,:),
(), INTENT(OUT) ::
INTEGER, INTENT(IN), OPTIONAL :: KL
(), INTENT(INOUT), OPTIONAL :: AFB(:,:)
INTEGER, INTENT(INOUT), OPTIONAL :: IPIV(:)
CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: TRANS, FACT
CHARACTER(LEN=1), INTENT(INOUT), OPTIONAL :: EQUED
REAL(), INTENT(INOUT), OPTIONAL :: C(:), R(:)
REAL(), INTENT(OUT), OPTIONAL :: , RCOND, RPVGRW
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 rectangular array, shape (:,:)
with size(AB,1) = kl + ku + 1 and size(AB,2) = n, where kl
and ku are, respectively, the numbers of subdiagonals and
superdiagonals in the band of A, and n is the order of A.
On entry, the matrix A or its equilibration in band storage.
The (kl + ku + 1) diagonals of A are stored in rows 1 to
(kl + ku + 1) of AB, so that the j-th column of A is
stored in the j-th column of AB as follows:
AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl)
1<=j<=n.
The remaining elements in AB need not be set.
If FACT = 'F' and EQUED /= 'N' then A has been equilibrated
by the scaling factors in R and/or C during the previous call
to LA GBSVX.
On exit, if FACT = 'E', 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 .
KL Optional (input) INTEGER.
The number of subdiagonals in the band of A (KL = kl).
The number of superdiagonals in the band is given by
ku = size(AB,1) - kl - 1.
Default value: (size(AB,1) - 1) / 2.
AFB Optional (input or output) REAL or COMPLEX rectangular array,
shape (:,:) with size(AFB,1) = 2*kl+ku+1 and size(AFB,2)=n
If FACT = 'F' then AFB is an input argument that contains the
details of the factorization of (the equilibrated) A returned
by a previous call to LA_GBSVX.
If FACT /= 'F' then AFB is an output argument that contains
the details of the factorization of (the equilibrated) A. U is
an upper triangular band matrix with (kl + ku + 1) diagonals.
These are stored in the first (kl + ku + 1) rows of AFB. The
multipliers that arise during the factorization are stored in
the remaining rows.
IPIV Optional (input or output) INTEGER array, shape (:) with
size(IPIV) = n.
If FACT = 'F' then IPIV is an input argument that contains
the pivot indices from the factorization of (the equilibrated)
A, returned by a previous call to LA_GBSVX.
If FACT /= 'F' then IPIV is an output argument that contains
the pivot indices from the factorization of (the equilibrated)
A.
FACT Optional (input) CHARACTER(LEN=1).
Specifies whether 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.
= '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 and IPIV contain the factored form of (the
equilibrated) A.
Default value: 'N'.
TRANS Optional (input) CHARACTER(LEN=1).
Specifies the form of the system of equations:
= 'N': A*X = B (No transpose)
= 'T': A^T*X = B (Transpose)
= 'C': A^H*X = B (Conjugate transpose)
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').
= 'R': Row equilibration, i.e., A has been premultiplied
by diag(R).
= 'C': Column equilibration, i.e., A has been postmultiplied
by diag(C).
= 'B': Both row and column equilibration.
Default value: 'N'.
R Optional (input or output) REAL array, shape (:) with
size(R) = size(A,1).
The row scale factors for A.
R is an input argument if FACT = 'F' and EQUED = 'R' or 'B'.
R is an output argument if FACT = 'E' and EQUED = 'R' or 'B'.
C Optional (input or output) REAL array, shape (:) with
size(C) = size(A,1).
The column scale factors for A.
C is an input argument if FACT = 'F' and EQUED = 'C' or 'B'.
C is an output argument if FACT = 'E' and EQUED = 'C' or 'B'.
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.
RPVGRW Optional (output) REAL.
The reciprocal pivot growth factor ||A||inf = ||U||inf. If
RPVGRW is much less than 1, then the stability of the LU
factorization of the (equilibrated) matrix A could be poor.
This also means that the solution X , condition estimator
RCOND, and forward error bound FERR could be unreliable. If
the factorization fails with 0 < INFO <= size(A,1), then
RPVGRW contains the reciprocal pivot growth factor for the
leading INFO columns of A.
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: U(i,i) = 0. The factorization has been completed,
but the factor U is singular, so the solution could
not be computed.
= n+1: U 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.