f04 Chapter Contents
f04 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_real_cholesky_solve_mult_rhs (f04agc)

## 1  Purpose

nag_real_cholesky_solve_mult_rhs (f04agc) calculates the approximate solution of a set of real symmetric positive definite linear equations with multiple right-hand sides, $AX=B$, where $A$ has been factorized by nag_real_cholesky (f03aec).

## 2  Specification

 #include #include
 void nag_real_cholesky_solve_mult_rhs (Integer n, Integer nrhs, double a[], Integer tda, double p[], const double b[], Integer tdb, double x[], Integer tdx, NagError *fail)

## 3  Description

To solve a set of real linear equations $AX=B$ where $A$ is symmetric positive definite, nag_real_cholesky_solve_mult_rhs (f04agc) must be preceded by a call to nag_real_cholesky (f03aec) which computes a Cholesky factorization of $A$ as $A={LL}^{\mathrm{T}}$, where $L$ is lower triangular. The columns $x$ of the solution $X$ are found by forward and backward substitution in $Ly=b$ and ${L}^{\mathrm{T}}x=y$, where $b$ is a column of the right-hand sides.

## 4  References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2:     nrhsIntegerInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 1$.
3:     a[${\mathbf{n}}×{\mathbf{tda}}$]doubleInput
Note: the $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{tda}}+j-1\right]$.
On entry: the upper triangle of the $n$ by $n$ positive definite symmetric matrix $A$, and the sub-diagonal elements of its Cholesky factor $L$, as returned by nag_real_cholesky (f03aec).
4:     tdaIntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: ${\mathbf{tda}}\ge {\mathbf{n}}$.
5:     p[n]doubleInput
On entry: the reciprocals of the diagonal elements of $L$, as returned by nag_real_cholesky (f03aec).
6:     b[${\mathbf{n}}×{\mathbf{tdb}}$]const doubleInput
Note: the $\left(i,j\right)$th element of the matrix $B$ is stored in ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{tdb}}+j-1\right]$.
On entry: the $n$ by $r$ right-hand side matrix $B$. See also Section 8.
7:     tdbIntegerInput
On entry: the stride separating matrix column elements in the array b.
Constraint: ${\mathbf{tdb}}\ge {\mathbf{nrhs}}$.
8:     x[${\mathbf{n}}×{\mathbf{tdx}}$]doubleOutput
Note: the $\left(i,j\right)$th element of the matrix $X$ is stored in ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{tdx}}+j-1\right]$.
On exit: the $n$ by $r$ solution matrix $X$. See also Section 8.
9:     tdxIntegerInput
On entry: the stride separating matrix column elements in the array x.
Constraint: ${\mathbf{tdx}}\ge {\mathbf{nrhs}}$.
10:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, ${\mathbf{tda}}=〈\mathit{\text{value}}〉$ while ${\mathbf{n}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tda}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{tdb}}=〈\mathit{\text{value}}〉$ while ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tdb}}\ge {\mathbf{nrhs}}$.
On entry, ${\mathbf{tdx}}=〈\mathit{\text{value}}〉$ while ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tdx}}\ge {\mathbf{nrhs}}$.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrhs}}\ge 1$.

## 7  Accuracy

The accuracy of the computed solutions depends on the conditioning of the original matrix. For a detailed error analysis see page 39 of Wilkinson and Reinsch (1971).

The time taken by nag_real_cholesky_solve_mult_rhs (f04agc) is approximately proportional to ${n}^{2}r$.
The function may be called with the same actual array supplied for arguments b and x, in which case the solution vectors will overwrite the right-hand sides.

## 9  Example

This example solves the set of linear equations $AX=B$ where
 $A = 5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10 and B = 23 32 33 31 .$

### 9.1  Program Text

Program Text (f04agce.c)

### 9.2  Program Data

Program Data (f04agce.d)

### 9.3  Program Results

Program Results (f04agce.r)