# NAG FL Interfacef07caf (dgtsv)

## 1Purpose

f07caf 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.

## 2Specification

Fortran Interface
 Subroutine f07caf ( n, nrhs, dl, d, du, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: dl(*), d(*), du(*), b(ldb,*)
#include <nag.h>
 void f07caf_ (const Integer *n, const Integer *nrhs, double dl[], double d[], double du[], double b[], const Integer *ldb, Integer *info)
The routine may be called by the names f07caf, nagf_lapacklin_dgtsv or its LAPACK name dgtsv.

## 3Description

f07caf uses Gaussian elimination with partial pivoting and row interchanges to solve the equations $AX=B$. The matrix $A$ is factorized as $A=PLU$, where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element per column, and $U$ is an upper triangular band matrix, with two superdiagonals.
Note that equations ${A}^{\mathrm{T}}X=B$ may be solved by interchanging the order of the arguments du and dl.

## 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 https://www.netlib.org/lapack/lug

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
3: $\mathbf{dl}\left(*\right)$Real (Kind=nag_wp) array Input/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 no constraints are violated, dl is overwritten by the ($n-2$) elements of the second superdiagonal of the upper triangular matrix $U$ from the $LU$ factorization of $A$, in ${\mathbf{dl}}\left(1\right),{\mathbf{dl}}\left(2\right),\dots ,{\mathbf{dl}}\left(n-2\right)$.
4: $\mathbf{d}\left(*\right)$Real (Kind=nag_wp) array Input/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 no constraints are violated, d is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
5: $\mathbf{du}\left(*\right)$Real (Kind=nag_wp) array Input/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 no constraints are violated, du is overwritten by the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
6: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
to solve the equations $Ax=b$, where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length ${\mathbf{ldb}}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
7: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07caf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero, and the solution has not been computed. The factorization has not been completed unless ${\mathbf{n}}=〈\mathit{\text{value}}〉$.

## 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
 $x^-x 1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Alternatives to f07caf, which return condition and error estimates are f04bcf and f07cbf.

## 8Parallelism and Performance

f07caf is not threaded in any implementation.

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$.
The complex analogue of this routine is f07cnf.

## 10Example

This example solves the equations
 $Ax=b ,$
where $A$ is the tridiagonal matrix
 $A = 3.0 2.1 0.0 0.0 0.0 3.4 2.3 -1.0 0.0 0.0 0.0 3.6 -5.0 1.9 0.0 0.0 0.0 7.0 -0.9 8.0 0.0 0.0 0.0 -6.0 7.1 and b = 2.7 -0.5 2.6 0.6 2.7 .$

### 10.1Program Text

Program Text (f07cafe.f90)

### 10.2Program Data

Program Data (f07cafe.d)

### 10.3Program Results

Program Results (f07cafe.r)