F04 Chapter Contents
F04 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF04BCF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F04BCF computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

## 2  Specification

 SUBROUTINE F04BCF ( N, NRHS, DL, D, DU, DU2, IPIV, B, LDB, RCOND, ERRBND, IFAIL)
 INTEGER N, NRHS, IPIV(N), LDB, IFAIL REAL (KIND=nag_wp) DL(*), D(*), DU(*), DU2(N-2), B(LDB,*), RCOND, ERRBND

## 3  Description

The $LU$ decomposition with partial pivoting and row interchanges is used to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element, and $U$ is an upper triangular band matrix with two superdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.
Note that the equations ${A}^{\mathrm{T}}X=B$ may be solved by interchanging the order of the arguments DU and DL.

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## 5  Parameters

1:     N – INTEGERInput
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
2:     NRHS – INTEGERInput
On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
3:     DL($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array DL must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
On exit: if ${\mathbf{IFAIL}}\ge {\mathbf{0}}$, DL is overwritten by the $\left(n-1\right)$ multipliers that define the matrix $L$ from the $LU$ factorization of $A$.
4:     D($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array D must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: must contain the $n$ diagonal elements of the matrix $A$.
On exit: if ${\mathbf{IFAIL}}\ge {\mathbf{0}}$, D is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
5:     DU($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension of the array DU must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ superdiagonal elements of the matrix $A$
On exit: if ${\mathbf{IFAIL}}\ge {\mathbf{0}}$, DU is overwritten by the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
6:     DU2(${\mathbf{N}}-2$) – REAL (KIND=nag_wp) arrayOutput
On exit: if ${\mathbf{IFAIL}}\ge {\mathbf{0}}$, DU2 returns the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
7:     IPIV(N) – INTEGER arrayOutput
On exit: if ${\mathbf{IFAIL}}\ge {\mathbf{0}}$, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row ${\mathbf{IPIV}}\left(i\right)$. ${\mathbf{IPIV}}\left(i\right)$ will always be either $i$ or $\left(i+1\right)$; ${\mathbf{IPIV}}\left(i\right)=i$ indicates a row interchange was not required.
8:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $n$ by $r$ matrix of right-hand sides $B$.
On exit: if ${\mathbf{IFAIL}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
9:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F04BCF is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
10:   RCOND – REAL (KIND=nag_wp)Output
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{RCOND}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
11:   ERRBND – REAL (KIND=nag_wp)Output
On exit: if ${\mathbf{IFAIL}}={\mathbf{0}}$ or $\mathbf{N}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{ERRBND}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array B and $x$ is the corresponding column of the exact solution $X$. If RCOND is less than machine precision, then ERRBND is returned as unity.
12:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
If ${\mathbf{IFAIL}}=-i$, the $i$th argument had an illegal value.
${\mathbf{IFAIL}}=-999$
Allocation of memory failed. The integer allocatable memory required is N, and the real allocatable memory required is $2×{\mathbf{N}}$. In this case the factorization and the solution $X$ have been computed, but RCOND and ERRBND have not been computed.
If ${\mathbf{IFAIL}}=i$, ${u}_{ii}$ is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be computed.
${\mathbf{IFAIL}}={\mathbf{N}}+1$
RCOND is less than machine precision, so that the matrix $A$ is numerically singular. A solution to the equations $AX=B$ has nevertheless been computed.

## 7  Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. F04BCF uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate ERRBND. See Section 4.4 of Anderson et al. (1999) for further details.

The total number of floating point operations required to solve the equations $AX=B$ is proportional to $nr$. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The complex analogue of F04BCF is F04CCF.

## 9  Example

This example solves the equations
 $AX=B,$
where $A$ is the tridiagonal matrix
 $A= 3.0 2.1 0 0 0 3.4 2.3 -1.0 0 0 0 3.6 -5.0 1.9 0 0 0 7.0 -0.9 8.0 0 0 0 -6.0 7.1 and B= 2.7 6.6 -0.5 10.8 2.6 -3.2 0.6 -11.2 2.7 19.1 .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.

### 9.1  Program Text

Program Text (f04bcfe.f90)

### 9.2  Program Data

Program Data (f04bcfe.d)

### 9.3  Program Results

Program Results (f04bcfe.r)