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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dptsv (f07ja)

## Purpose

nag_lapack_dptsv (f07ja) computes the solution to a real system of linear equations
 AX = B , $AX=B ,$
where A$A$ is an n$n$ by n$n$ symmetric positive definite tridiagonal matrix, and X$X$ and B$B$ are n$n$ by r$r$ matrices.

## Syntax

[d, e, b, info] = f07ja(d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[d, e, b, info] = nag_lapack_dptsv(d, e, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dptsv (f07ja) factors A$A$ as A = LDLT$A=LD{L}^{\mathrm{T}}$. The factored form of A$A$ is then used to solve the system of equations.

## 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ diagonal elements of the tridiagonal matrix A$A$.
2:     e( : $:$) – double array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
The (n1)$\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix A$A$.
3:     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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
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 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:     d( : $:$) – double array
Note: the dimension of the array d must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ diagonal elements of the diagonal matrix D$D$ from the factorization A = LDLT$A=LD{L}^{\mathrm{T}}$.
2:     e( : $:$) – double array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
The (n1)$\left(n-1\right)$ subdiagonal elements of the unit bidiagonal factor L$L$ from the LDLT$LD{L}^{\mathrm{T}}$ factorization of A$A$. (e can also be regarded as the superdiagonal of the unit bidiagonal factor U$U$ from the UTDU${U}^{\mathrm{T}}DU$ factorization of A$A$.)
3:     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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
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$X$.
4:     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

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: d, 4: e, 5: b, 6: ldb, 7: 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.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the leading minor of order i$i$ is not positive definite, 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.
nag_lapack_dptsvx (f07jb) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_real_posdef_tridiag_solve (f04bg) solves Ax = b $Ax=b$ and returns a forward error bound and condition estimate. nag_linsys_real_posdef_tridiag_solve (f04bg) calls nag_lapack_dptsv (f07ja) to solve the equations.

The number of floating point operations required for the factorization of A $A$ is proportional to n $n$, and the number of floating point operations required for the solution of the equations is proportional to nr $nr$, where r $r$ is the number of right-hand sides.
The complex analogue of this function is nag_lapack_zptsv (f07jn).

## Example

```function nag_lapack_dptsv_example
d = [4;
10;
29;
25;
5];
e = [-2;
-6;
15;
8];
b = [6;
9;
2;
14;
7];
[dOut, eOut, bOut, info] = nag_lapack_dptsv(d, e, b)
```
```

dOut =

4
9
25
16
1

eOut =

-0.5000
-0.6667
0.6000
0.5000

bOut =

2.5000
2.0000
1.0000
-1.0000
3.0000

info =

0

```
```function f07ja_example
d = [4;
10;
29;
25;
5];
e = [-2;
-6;
15;
8];
b = [6;
9;
2;
14;
7];
[dOut, eOut, bOut, info] = f07ja(d, e, b)
```
```

dOut =

4
9
25
16
1

eOut =

-0.5000
-0.6667
0.6000
0.5000

bOut =

2.5000
2.0000
1.0000
-1.0000
3.0000

info =

0

```