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NAG Toolbox: nag_lapack_dgttrs (f07ce)

Purpose

nag_lapack_dgttrs (f07ce) computes the solution to a real system of linear equations AX = B $AX=B$ or ATX = B ${A}^{\mathrm{T}}X=B$, 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, using the LU $LU$ factorization returned by nag_lapack_dgttrf (f07cd).

Syntax

[b, info] = f07ce(trans, dl, d, du, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dgttrs(trans, dl, d, du, du2, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgttrs (f07ce) should be preceded by a call to nag_lapack_dgttrf (f07cd), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix A $A$ 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 in each column, and U $U$ is an upper triangular band matrix, with two superdiagonals. nag_lapack_dgttrs (f07ce) then utilizes the factorization to solve the required 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

Parameters

Compulsory Input Parameters

1:     trans – string (length ≥ 1)
Specifies the equations to be solved as follows:
trans = 'N'${\mathbf{trans}}=\text{'N'}$
Solve AX = B$AX=B$ for X$X$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$
Solve ATX = B${A}^{\mathrm{T}}X=B$ for X$X$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$, 'T'$\text{'T'}$ or 'C'$\text{'C'}$.
2:     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)$\left(n-1\right)$ multipliers that define the matrix L$L$ of the LU$LU$ factorization of A$A$.
3:     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 upper triangular matrix U$U$ from the LU$LU$ factorization of A$A$.
4:     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)$\left(n-1\right)$ elements of the first superdiagonal of U$U$.
5:     du2( : $:$) – double array
Note: the dimension of the array du2 must be at least max (1,n2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-2\right)$.
Must contain the (n2)$\left(n-2\right)$ elements of the second superdiagonal of U$U$.
6:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Must contain the n$n$ pivot indices that define the permutation matrix P$P$. At the i$i$th step, row i$i$ of the matrix was interchanged with row ipiv(i)${\mathbf{ipiv}}\left(i\right)$, and ipiv(i)${\mathbf{ipiv}}\left(i\right)$ must always be either i$i$ or (i + 1)$\left(i+1\right)$, ipiv(i) = i${\mathbf{ipiv}}\left(i\right)=i$ indicating that a row interchange was not performed.
7:     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$r$ 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 arrays d, ipiv.
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, : $:$) – 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)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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: trans, 2: n, 3: nrhs_p, 4: dl, 5: d, 6: du, 7: du2, 8: ipiv, 9: b, 10: ldb, 11: 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_dgtcon (f07cg) can be used to estimate the condition number of A $A$ and nag_lapack_dgtrfs (f07ch) can be used to obtain approximate error bounds.

Further Comments

The total number of floating point operations required to solve the equations AX = B $AX=B$ or ATX = B ${A}^{\mathrm{T}}X=B$ is proportional to nr $nr$.
The complex analogue of this function is nag_lapack_zgttrs (f07cs).

Example

```function nag_lapack_dgttrs_example
trans = 'No transpose';
dl = [0.8823529411764706;
0.01960784313725495;
0.1400560224089636;
-0.01479925303454714];
d = [3.4;
3.6;
7;
-6;
-1.015373482726424];
du = [2.3;
-5;
-0.9;
7.1];
du2 = [-1;
1.9;
8];
ipiv = [int64(2);3;4;5;5];
b = [2.7, 6.6;
-0.5, 10.8;
2.6, -3.2;
0.6, -11.2;
2.7, 19.1];
[bOut, info] = nag_lapack_dgttrs(trans, dl, d, du, du2, ipiv, b)
```
```

bOut =

-4.0000    5.0000
7.0000   -4.0000
3.0000   -3.0000
-4.0000   -2.0000
-3.0000    1.0000

info =

0

```
```function f07ce_example
trans = 'No transpose';
dl = [0.8823529411764706;
0.01960784313725495;
0.1400560224089636;
-0.01479925303454714];
d = [3.4;
3.6;
7;
-6;
-1.015373482726424];
du = [2.3;
-5;
-0.9;
7.1];
du2 = [-1;
1.9;
8];
ipiv = [int64(2);3;4;5;5];
b = [2.7, 6.6;
-0.5, 10.8;
2.6, -3.2;
0.6, -11.2;
2.7, 19.1];
[bOut, info] = f07ce(trans, dl, d, du, du2, ipiv, b)
```
```

bOut =

-4.0000    5.0000
7.0000   -4.0000
3.0000   -3.0000
-4.0000   -2.0000
-3.0000    1.0000

info =

0

```

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