f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_superlu_solve_lu (f11mfc)

## 1  Purpose

nag_superlu_solve_lu (f11mfc) solves a real sparse system of linear equations with multiple right-hand sides given an $LU$ factorization of the sparse matrix computed by nag_superlu_lu_factorize (f11mec).

## 2  Specification

 #include #include
 void nag_superlu_solve_lu (Nag_OrderType order, Nag_TransType trans, Integer n, const Integer iprm[], const Integer il[], const double lval[], const Integer iu[], const double uval[], Integer nrhs, double b[], Integer pdb, NagError *fail)

## 3  Description

nag_superlu_solve_lu (f11mfc) solves a real system of linear equations with multiple right-hand sides $AX=B$ or ${A}^{\mathrm{T}}X=B$, according to the value of the argument trans, where the matrix factorization ${P}_{r}A{P}_{c}=LU$ corresponds to an $LU$ decomposition of a sparse matrix stored in compressed column (Harwell–Boeing) format, as computed by nag_superlu_lu_factorize (f11mec).
In the above decomposition $L$ is a lower triangular sparse matrix with unit diagonal elements and $U$ is an upper triangular sparse matrix; ${P}_{r}$ and ${P}_{c}$ are permutation matrices.

None.

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     transNag_TransTypeInput
On entry: specifies whether $AX=B$ or ${A}^{\mathrm{T}}X=B$ is solved.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$AX=B$ is solved.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
${A}^{\mathrm{T}}X=B$ is solved.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     iprm[$7×{\mathbf{n}}$]const IntegerInput
On entry: the column permutation which defines ${P}_{c}$, the row permutation which defines ${P}_{r}$, plus associated data structures as computed by nag_superlu_lu_factorize (f11mec).
5:     il[$\mathit{dim}$]const IntegerInput
Note: the dimension, dim, of the array il must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).
On entry: records the sparsity pattern of matrix $L$ as computed by nag_superlu_lu_factorize (f11mec).
6:     lval[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array lval must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).
On entry: records the nonzero values of matrix $L$ and some nonzero values of matrix $U$ as computed by nag_superlu_lu_factorize (f11mec).
7:     iu[$\mathit{dim}$]const IntegerInput
Note: the dimension, dim, of the array iu must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).
On entry: records the sparsity pattern of matrix $U$ as computed by nag_superlu_lu_factorize (f11mec).
8:     uval[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array uval must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).
On entry: records some nonzero values of matrix $U$ as computed by nag_superlu_lu_factorize (f11mec).
9:     nrhsIntegerInput
On entry: $\mathit{nrhs}$, the number of right-hand sides in $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
10:   b[$\mathit{dim}$]doubleInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the ${\mathbf{n}}$ by ${\mathbf{nrhs}}$ right-hand side matrix $B$.
On exit: the ${\mathbf{n}}$ by ${\mathbf{nrhs}}$ solution matrix $X$.
11:   pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument number $〈\mathit{\text{value}}〉$ had an illegal value.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_INVALID_PERM_COL
Incorrect Column Permutations in array iprm.
NE_INVALID_PERM_ROW
Incorrect Row Permutations in array iprm.

## 7  Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
 $E≤cnεLU,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision, when partial pivoting is used.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤cncondA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$, and $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ can be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_superlu_refine_lu (f11mhc), and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling nag_superlu_condition_number_lu (f11mgc).

nag_superlu_solve_lu (f11mfc) may be followed by a call to nag_superlu_refine_lu (f11mhc) to refine the solution and return an error estimate.

## 9  Example

This example solves the system of equations $AX=B$, where
 $A= 2.00 1.00 0 0 0 0 0 1.00 -1.00 0 4.00 0 1.00 0 1.00 0 0 0 1.00 2.00 0 -2.00 0 0 3.00 and B= 1.56 3.12 -0.25 -0.50 3.60 7.20 1.33 2.66 0.52 1.04 .$
Here $A$ is nonsymmetric and must first be factorized by nag_superlu_lu_factorize (f11mec).

### 9.1  Program Text

Program Text (f11mfce.c)

### 9.2  Program Data

Program Data (f11mfce.d)

### 9.3  Program Results

Program Results (f11mfce.r)