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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpttrs (f07js)

## Purpose

nag_lapack_zpttrs (f07js) computes the solution to a complex system of linear equations AX = B $AX=B$, where A $A$ is an n $n$ by n $n$ Hermitian positive definite tridiagonal matrix and X $X$ and B $B$ are n $n$ by r $r$ matrices, using the LDLH $LD{L}^{\mathrm{H}}$ factorization returned by nag_lapack_zpttrf (f07jr).

## Syntax

[b, info] = f07js(uplo, d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpttrs(uplo, d, e, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zpttrs (f07js) should be preceded by a call to nag_lapack_zpttrf (f07jr), which computes a modified Cholesky factorization of the matrix A $A$ as
 A = LDLH , $A=LDLH ,$
where L $L$ is a unit lower bidiagonal matrix and D $D$ is a diagonal matrix, with positive diagonal elements. nag_lapack_zpttrs (f07js) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form UHDU ${U}^{\mathrm{H}}DU$, where U $U$ is a unit upper bidiagonal matrix.

## 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:     uplo – string (length ≥ 1)
Specifies the form of the factorization as follows:
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = UHDU$A={U}^{\mathrm{H}}DU$.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = LDLH$A=LD{L}^{\mathrm{H}}$.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
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 diagonal matrix D$D$ from the LDLH$LD{L}^{\mathrm{H}}$ or UHDU${U}^{\mathrm{H}}DU$ factorization of A$A$.
3:     e( : $:$) – complex array
Note: the dimension of the array e must be at least max (1,n1)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, e must contain the (n1)$\left(n-1\right)$ superdiagonal elements of the unit upper bidiagonal matrix U$U$ from the UHDU${U}^{\mathrm{H}}DU$ factorization of A$A$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, e must contain the (n1)$\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix L$L$ from the LDLH$LD{L}^{\mathrm{H}}$ factorization of A$A$.
4:     b(ldb, : $:$) – complex 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$r$ matrix of right-hand sides 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:     b(ldb, : $:$) – complex 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)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ solution matrix X$X$.
2:     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: uplo, 2: n, 3: nrhs_p, 4: d, 5: e, 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.

## 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.
Following the use of this function nag_lapack_zptcon (f07ju) can be used to estimate the condition number of A $A$ and nag_lapack_zptrfs (f07jv) can be used to obtain approximate error bounds.

The total number of floating point operations required to solve the equations AX = B $AX=B$ is proportional to nr $nr$.
The real analogue of this function is nag_lapack_dpttrs (f07je).

## Example

```function nag_lapack_zpttrs_example
uplo = 'U';
d = [16;
9;
1;
4];
e = [ 1 - 1i;
2 + 1i;
1 + 4i];
b = [ 64 + 16i,  -16 - 32i;
93 + 62i,  61 - 66i;
78 - 80i,  71 - 74i;
14 - 27i,  35 + 15i];
[bOut, info] = nag_lapack_zpttrs(uplo, d, e, b)
```
```

bOut =

2.0000 + 1.0000i  -3.0000 - 2.0000i
1.0000 + 1.0000i   1.0000 + 1.0000i
1.0000 - 2.0000i   1.0000 - 2.0000i
1.0000 - 1.0000i   2.0000 + 1.0000i

info =

0

```
```function f07js_example
uplo = 'U';
d = [16;
9;
1;
4];
e = [ 1 - 1i;
2 + 1i;
1 + 4i];
b = [ 64 + 16i,  -16 - 32i;
93 + 62i,  61 - 66i;
78 - 80i,  71 - 74i;
14 - 27i,  35 + 15i];
[bOut, info] = f07js(uplo, d, e, b)
```
```

bOut =

2.0000 + 1.0000i  -3.0000 - 2.0000i
1.0000 + 1.0000i   1.0000 + 1.0000i
1.0000 - 2.0000i   1.0000 - 2.0000i
1.0000 - 1.0000i   2.0000 + 1.0000i

info =

0

```