Arguments

AB

(input/output) REAL or COMPLEX rectangular array, shape $(:,:)$ with $size({\bf AB},1) = kl+ku+1$ and $size({\bf AB},2) = n$, 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:

$$\begin{array}{c\vert c} A_{i,j} & i,j \\ \hline {\bf AB}_{k... ...(n,j+kl) \\ 1 \leq j \leq n \end{array} \\ \hline \end{array}$$

The remaining elements in ${\bf AB}$ need not be set.
If ${\bf FACT} =$ 'F' and ${\bf EQUED} \neq$ 'N' then $A$ has been equilibrated by the scaling factors in ${\bf R}$ and/or ${\bf C}$ during the previous call to LA_GBSVX.
On exit, if ${\bf FACT} =$ 'E', the equilibrated version of $A$ is stored in AB; otherwise, ${\bf AB}$ is unchanged.

B

(input/output) REAL or COMPLEX array, shape $(:,:)$ with $size({\bf B},1) = n$ or shape $(:)$ with $size({\bf B}) = n$.
On entry, the matrix $B$.
On exit, the scaled version of $B$ if the system has been equilibrated; otherwise, ${\bf B}$ is unchanged.

X

(output) REAL or COMPLEX array, shape $(:,:)$ with $size({\bf X},1) = n$ and $size({\bf X},2) = size({\bf B},2)$, or shape $(:)$ with $size({\bf X})=n$.
The solution matrix $X$.

KL

Optional (input) INTEGER.
The number of subdiagonals in the band of $A$ (${\bf KL} = kl$).
The number of superdiagonals in the band is given by $ku = size({\bf AB},1) - kl -1$.
Default value: $(size({\bf AB},1)-1)/2$.

AFB

Optional (input or output) REAL or COMPLEX rectangular array, shape $(:,:)$ with $size({\bf AFB},1) = 2 kl+ku+1$ and $size({\bf AFB},2) = n$
If ${\bf FACT}$ = '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 ${\bf FACT} \neq$ 'F' then ${\bf AFB}$ 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 ${\bf AFB}$. The multipliers that arise during the factorization are stored in the remaining rows.

IPIV

Optional (input or output) INTEGER array, shape $(:)$ with $size({\bf IPIV})=n$.
If ${\bf FACT}$ = 'F' then ${\bf IPIV}$ 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 ${\bf FACT} \neq$ 'F' then ${\bf IPIV}$ 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.


Default value: 'N'.

TRANS

Optional (input) CHARACTER(LEN=1).
Specifies the form of the system of equations:


Default value: 'N'.

EQUED

Optional (input or output) CHARACTER(LEN=1).
Specifies the form of equilibration that was done.
${\bf EQUED}$ is an input argument if ${\bf FACT} =$ 'F', otherwise it is an output argument:


Default value: 'N'.

R

Optional (input or output) REAL array, shape $(:)$ with $size({\bf R})=size({\bf A},1)$.
The row scale factors for $A$.
R is an input argument if FACT = 'F' and ${\bf EQUED}=$ 'R' or 'B'.
R is an output argument if FACT = 'E' and ${\bf EQUED}=$ 'R' or 'B'.

C

Optional (input or output) REAL array, shape $(:)$ with $size({\bf C})=size({\bf A},1)$.
The column scale factors for $A$.
C is an input argument if FACT = 'F' and ${\bf EQUED}=$ 'C' or 'B'.
C is an output argument if FACT = 'E' and ${\bf EQUED}=$ 'C' or 'B'.

FERR

Optional (output) REAL array of shape $(:)$, with $size({\bf FERR})=size({\bf X},2)$, 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 $X_j$, ${\bf FERR}_j$ 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 $size({\bf BERR})=size({\bf X},2)$, 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 $\Vert A\Vert _{\infty}/\Vert U\Vert _{\infty}$. 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 ${\bf FERR}$ could be unreliable. If the factorization fails with $0<{\bf INFO}\leq size({\bf A},1)$, then RPVGRW contains the reciprocal pivot growth factor for the leading INFO columns of $A$.

INFO

Optional (output) INTEGER


If ${\bf INFO}$ is not present and an error occurs, then the program is terminated with an error message.

References: [1] and [17,9,20,21].