# NAG Library Routine Document

## 1Purpose

f04bgf computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ symmetric positive definite 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.

## 2Specification

Fortran Interface
 Subroutine f04bgf ( n, nrhs, d, e, b, ldb,
 Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: d(*), e(*), b(ldb,*) Real (Kind=nag_wp), Intent (Out) :: rcond, errbnd
C Header Interface
#include nagmk26.h
 void f04bgf_ (const Integer *n, const Integer *nrhs, double d[], double e[], double b[], const Integer *ldb, double *rcond, double *errbnd, Integer *ifail)

## 3Description

$A$ is factorized as $A=LD{L}^{\mathrm{T}}$, where $L$ is a unit lower bidiagonal matrix and $D$ is diagonal, and the factored form of $A$ is then used to solve the system of equations.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{nrhs}$ – 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:     $\mathbf{d}\left(*\right)$ – 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 tridiagonal matrix $A$.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, d is overwritten by the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
4:     $\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array e 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 tridiagonal matrix $A$.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, e is overwritten by the $\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix $L$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$. (e can also be regarded as the superdiagonal of the unit upper bidiagonal factor $U$ from the ${U}^{\mathrm{T}}DU$ factorization of $A$.)
5:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – 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 $X$.
6:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f04bgf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     $\mathbf{rcond}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
8:     $\mathbf{errbnd}$ – 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, errbnd is returned as unity.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{n}}$
The principal minor of order $〈\mathit{\text{value}}〉$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.
${\mathbf{ifail}}={\mathbf{n}}+1$
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
${\mathbf{ifail}}=-6$
On entry, ${\mathbf{ldb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
The real allocatable memory required is n. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been computed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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. f04bgf uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f04bgf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$. The condition number estimation requires $\mathit{O}\left(n\right)$ floating-point operations.
See Section 15.3 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.
The complex analogue of f04bgf is f04cgf.

## 10Example

This example solves the equations
 $AX=B,$
where $A$ is the symmetric positive definite tridiagonal matrix
 $A= 4.0 -2.0 0 0 0 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0 and B= 6.0 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.

### 10.1Program Text

Program Text (f04bgfe.f90)

### 10.2Program Data

Program Data (f04bgfe.d)

### 10.3Program Results

Program Results (f04bgfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017