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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgtsv (f07ca)

## Purpose

nag_lapack_dgtsv (f07ca) computes the solution to a real system of linear equations
 AX = B , $AX=B ,$
where A$A$ is an n$n$ by n$n$ tridiagonal matrix and X$X$ and B$B$ are n$n$ by r$r$ matrices.

## Syntax

[dl, d, du, b, info] = f07ca(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)
[dl, d, du, b, info] = nag_lapack_dgtsv(dl, d, du, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgtsv (f07ca) uses Gaussian elimination with partial pivoting and row interchanges to solve the equations AX = B $AX=B$. The matrix A $A$ is factorized as A = PLU $A=PLU$, where P $P$ is a permutation matrix, L $L$ is unit lower triangular with at most one nonzero subdiagonal element per column, and U $U$ is an upper triangular band matrix, with two superdiagonals.
Note that the equations ATX = B${A}^{\mathrm{T}}X=B$ may be solved by interchanging the order of the arguments du and dl.

## 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

## Parameters

### Compulsory Input Parameters

1:     dl( : $:$) – double array
Note: the dimension of the array dl must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ subdiagonal elements of the matrix A$A$.
2:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ diagonal elements of the matrix A$A$.
3:     du( : $:$) – double array
Note: the dimension of the array du must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
Must contain the (n1)$\left(n-1\right)$ superdiagonal elements of the matrix A$A$.
4:     b(ldb, : $:$) – double array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
Note: to solve the equations Ax = b$Ax=b$, where b$b$ is a single right-hand side, b may be supplied as a one-dimensional array with length ldb = max (1,n)$\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ right-hand side matrix B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b The dimension of the array d.
n$n$, the number of linear equations, i.e., the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
r$r$, the number of right-hand sides, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldb

### Output Parameters

1:     dl( : $:$) – double array
Note: the dimension of the array dl must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If no constrains are violated, dl stores the (n2$n-2$) elements of the second superdiagonal of the upper triangular matrix U$U$ from the LU$LU$ factorization of A$A$, in dl(1),dl(2),,dl(n2)${\mathbf{dl}}\left(1\right),{\mathbf{dl}}\left(2\right),\dots ,{\mathbf{dl}}\left(n-2\right)$.
2:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If no constraints are violated, d stores the n$n$ diagonal elements of the upper triangular matrix U$U$ from the LU$LU$ factorization of A$A$.
3:     du( : $:$) – double array
Note: the dimension of the array du must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If no constraints are violated, du stores the (n1)$\left(n-1\right)$ elements of the first superdiagonal of U$U$.
4:     b(ldb, : $:$) – double array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
Note: to solve the equations Ax = b$Ax=b$, where b$b$ is a single right-hand side, b may be supplied as a one-dimensional array with length ldb = max (1,n)$\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{INFO}}={\mathbf{0}}$, the n$n$ by r$r$ solution matrix X$X$.
5:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: nrhs_p, 3: dl, 4: d, 5: du, 6: b, 7: ldb, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, uii${u}_{ii}$ is exactly zero, and the solution has not been computed. The factorization has not been completed unless i = n$i={\mathbf{n}}$.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 (A + E) x̂ = b , $(A+E) x^ = b ,$
where
 ‖E‖1 = O(ε) ‖A‖1 $‖E‖1 = O(ε) ‖A‖1$
and ε $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖1)/(‖x‖1) ≤ κ(A) (‖E‖1)/(‖A‖1) , $‖ x^-x ‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where κ(A) = A11 A1 $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Alternatives to nag_lapack_dgtsv (f07ca), which return condition and error estimates are nag_linsys_real_tridiag_solve (f04bc) and nag_lapack_dgtsvx (f07cb).

The total number of floating point operations required to solve the equations AX = B $AX=B$ is proportional to nr $nr$.
The complex analogue of this function is nag_lapack_zgtsv (f07cn).

## Example

```function nag_lapack_dgtsv_example
dl = [3.4;
3.6;
7;
-6];
d = [3;
2.3;
-5;
-0.9;
7.1];
du = [2.1;
-1;
1.9;
8];
b = [2.7;
-0.5;
2.6;
0.6;
2.7];
[dlOut, dOut, duOut, bOut, info] = nag_lapack_dgtsv(dl, d, du, b)
```
```

dlOut =

-1.0000
1.9000
8.0000
-6.0000

dOut =

3.4000
3.6000
7.0000
-6.0000
-1.0154

duOut =

2.3000
-5.0000
-0.9000
7.1000

bOut =

-4.0000
7.0000
3.0000
-4.0000
-3.0000

info =

0

```
```function f07ca_example
dl = [3.4;
3.6;
7;
-6];
d = [3;
2.3;
-5;
-0.9;
7.1];
du = [2.1;
-1;
1.9;
8];
b = [2.7;
-0.5;
2.6;
0.6;
2.7];
[dlOut, dOut, duOut, bOut, info] = f07ca(dl, d, du, b)
```
```

dlOut =

-1.0000
1.9000
8.0000
-6.0000

dOut =

3.4000
3.6000
7.0000
-6.0000
-1.0154

duOut =

2.3000
-5.0000
-0.9000
7.1000

bOut =

-4.0000
7.0000
3.0000
-4.0000
-3.0000

info =

0

```