Arguments

AB

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

$$\begin{array}{c\vert c\vert c} A_{i,j} & i,j & {\bf UPLO} ... ...1 \leq j \leq n \end{array} & \mbox{'L'} \\ \hline \end{array}$$

On exit, if ${\bf FACT} =$ 'E', then 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 $X$.

UPLO

Optional (input) CHARACTER(LEN=1).


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 $\neq$ '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.


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'.

S

Optional (input or output) REAL array, shape $(:)$ with $size(S) = size({\bf A},1)$.
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 $size({\bf FERR})=size({\bf X},2)$, 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 ${\bf RCOND}$, 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 ${\bf RCOND}$ is less than the machine precision, the matrix is singular to working precision. This condition is indicated by a return code of ${\bf INFO} > 0$.

INFO

Optional (output) INTEGER


If INFO is not present and an error occurs, then the program is terminated with an error message.

References: [1] and [17,9,20,21].