# NAG CL Interfacef11mhc (direct_​real_​gen_​refine)

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## 1Purpose

f11mhc returns error bounds for the solution of a real sparse system of linear equations with multiple right-hand sides, $AX=B$ or ${A}^{\mathrm{T}}X=B$. It improves the solution by iterative refinement in standard precision, in order to reduce the backward error as much as possible.

## 2Specification

 #include
 void f11mhc (Nag_OrderType order, Nag_TransType trans, Integer n, const Integer icolzp[], const Integer irowix[], const double a[], const Integer iprm[], const Integer il[], const double lval[], const Integer iu[], const double uval[], Integer nrhs, const double b[], Integer pdb, double x[], Integer pdx, double ferr[], double berr[], NagError *fail)
The function may be called by the names: f11mhc, nag_sparse_direct_real_gen_refine or nag_superlu_refine_lu.

## 3Description

f11mhc returns the backward errors and estimated bounds on the forward errors for the solution of a real system of linear equations with multiple right-hand sides $AX=B$ or ${A}^{\mathrm{T}}X=B$. The function handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of f11mhc in terms of a single right-hand side $b$ and solution $x$.
Given a computed solution $x$, the function computes the component-wise backward error $\beta$. This is the size of the smallest relative perturbation in each element of $A$ and $b$ such that if $x$ is the exact solution of a perturbed system:
 $(A+δA) x = b + δ b then |δaij| ≤ β |aij| and |δbi| ≤ β |bi| .$
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
 $maxi |xi-x^i| / maxi |xi|$
where $\stackrel{^}{x}$ is the true solution.
The function uses the $LU$ factorization ${P}_{r}A{P}_{c}=LU$ computed by f11mec and the solution computed by f11mfc.

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{trans}$Nag_TransType Input
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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{icolzp}\left[\mathit{dim}\right]$const Integer Input
Note: the dimension, dim, of the array icolzp must be at least ${\mathbf{n}}+1$.
On entry: the new column index array of sparse matrix $A$. See Section 2.1.3 in the F11 Chapter Introduction.
5: $\mathbf{irowix}\left[\mathit{dim}\right]$const Integer Input
Note: the dimension, dim, of the array irowix must be at least ${\mathbf{icolzp}}\left[{\mathbf{n}}\right]-1$, the number of nonzeros of the sparse matrix $A$.
On entry: the row index array of sparse matrix $A$. See Section 2.1.3 in the F11 Chapter Introduction.
6: $\mathbf{a}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array a must be at least ${\mathbf{icolzp}}\left[{\mathbf{n}}\right]-1$, the number of nonzeros of the sparse matrix $A$.
On entry: the array of nonzero values in the sparse matrix $A$.
7: $\mathbf{iprm}\left[7×{\mathbf{n}}\right]$const Integer Input
On entry: the column permutation which defines ${P}_{c}$, the row permutation which defines ${P}_{r}$, plus associated data structures as computed by f11mec.
8: $\mathbf{il}\left[\mathit{dim}\right]$const Integer Input
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 f11mec.
On entry: records the sparsity pattern of matrix $L$ as computed by f11mec.
9: $\mathbf{lval}\left[\mathit{dim}\right]$const double Input
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 f11mec.
On entry: records the nonzero values of matrix $L$ and some nonzero values of matrix $U$ as computed by f11mec.
10: $\mathbf{iu}\left[\mathit{dim}\right]$const Integer Input
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 f11mec.
On entry: records the sparsity pattern of matrix $U$ as computed by f11mec.
11: $\mathbf{uval}\left[\mathit{dim}\right]$const double Input
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 f11mec.
On entry: records some nonzero values of matrix $U$ as computed by f11mec.
12: $\mathbf{nrhs}$Integer Input
On entry: $\mathit{nrhs}$, the number of right-hand sides in $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
13: $\mathbf{b}\left[\mathit{dim}\right]$const double Input
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 $n×\mathit{nrhs}$ right-hand side matrix $B$.
14: $\mathbf{pdb}$Integer Input
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
15: $\mathbf{x}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdx}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $X$ is stored in
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n×\mathit{nrhs}$ solution matrix $X$, as returned by f11mfc.
On exit: the $n×\mathit{nrhs}$ improved solution matrix $X$.
16: $\mathbf{pdx}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
17: $\mathbf{ferr}\left[{\mathbf{nrhs}}\right]$double Output
On exit: ${\mathbf{ferr}}\left[\mathit{j}-1\right]$ contains an estimated error bound for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,\mathit{nrhs}$.
18: $\mathbf{berr}\left[{\mathbf{nrhs}}\right]$double Output
On exit: ${\mathbf{berr}}\left[\mathit{j}-1\right]$ contains the component-wise backward error bound $\beta$ for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,\mathit{nrhs}$.
19: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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: ${\mathbf{pdb}}>0$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}>0$.
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: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_PERM_COL
Incorrect column permutations in array iprm.
NE_INVALID_PERM_ROW
Incorrect row permutations in array iprm.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

## 8Parallelism and Performance

f11mhc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11mhc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form $Ax=b$ or ${A}^{\mathrm{T}}x=b$;

## 10Example

This example solves the system of equations $AX=B$ using iterative refinement and to compute the forward and backward error bounds, 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 f11mec.

### 10.1Program Text

Program Text (f11mhce.c)

### 10.2Program Data

Program Data (f11mhce.d)

### 10.3Program Results

Program Results (f11mhce.r)