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# solve_linsys

Computes the solution to a real system of linear equations.

## Prototype

```	function solve_linsys (
A [*][*] : numeric,
B        : numeric
)

return_val [dimsizes(B)] :  float or double
```

## Arguments

A

A two-dimensional (N, N) coefficient matrix.

B

A one- or two-dimensional array containing the right hand side matrix. If B is one-dimensional, it must be of length N. If it is two-dimensional, it must be B(NRHS, N) where NRHS is the number of right hand sides.

## Return value

Returns an array dimensioned the same as B.

The return type is floating point if the input is floating point, and double if the input is of type double.

## Description

This function computes the solution to a real system of linear equations A * x = B where B may have multiple right-hand-sides. An LU decomposition with partial pivoting and row interchange is used to factor A as A = P * L * U where P is a permutation matrix, L is the unit lower tridiagonalar matrix, and U is the upper triangular matrix. The factored form is then used to solve the system of equations.

## Examples

Example 1:

Solve the system A * x = B where B is one-dimensional:

```                 3  2 -1  2           0
A =   1  4  0  2       B = 0
2  1  2 -1           1
1  1 -1  3           0

```
In NCL:
```   A = (/ (/3.0, 2.0, -1.0, 2.0/), \
(/1.0, 4.0,  0.0, 2.0/), \
(/2.0, 1.0,  2.0,-1.0/), \
(/1.0, 1.0, -1.0, 3.0/) /)

B = (/0.0, 0.0, 1.0, 0.0/)  ; this is treated as a one-dimensional
; column vector (special case)

x = solve_linsys(A, B)      ; x will be one-dimensional [x(N)]
```
Note: the following would also work:
```   NRHS    = 1
N       = 4
B       = new ((/NRHS,N/), "float")   ; "double" also
B(0, 0) = 0.0
B(0, 1) = 0.0
B(0, 2) = 1.0
B(0, 3) = 0.0
```
Example 2:

The same as Example 1, but with multiple right hand sides:

```   nrhs   = 3
n      = 4
b      = new ((/nrhs,n/), "float")   ; "double" also
b(0,:) = (/0.0, 0.0, 1.0, 0.0/)
b(1,:) = (/-2.0, 1.0, 3.0, 4.0/)
b(2,:) = (/2.0, 2.0, 0.0, 0.0/)

x = solve_linsys(A, B)  ; x will be 2D: x(3, 4)
```

## Errors

info
= 0 : successful exit

< 0: the i-th argument had an illegal value

> 0: U(i, i) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.