# Use APDLMath to Solve a Dense Matrix Linear System¶

Use the APDLMath module to solve a Dense Matrix Linear System.

```import time
import numpy.linalg as np
from ansys.mapdl.core import launch_mapdl

# Start MAPDL as a service and create an APDLMath object.
mapdl = launch_mapdl()
mm = mapdl.math
```

Allocate a Dense Matrix in the APDLMath workspace

```mapdl.clear()
dim = 10000
a = mm.rand(dim, dim)
b = mm.rand(dim)
x = mm.zeros(dim)
```

Copy the matrices as numpy arrays before they are modified by factorization call

```a_py = a.asarray()
b_py = b.asarray()
```

Solve using APDLMath

```print(f'Solving a ({dim} x {dim}) dense linear system using MAPDL...')

t1 = time.time()
s = mm.factorize(a)
x = s.solve(b, x)
t2 = time.time()
print(f'Elapsed time to solve the linear system using Mapdl: {t2 - t1} seconds')
```

Out:

```Solving a (10000 x 10000) dense linear system using MAPDL...
Elapsed time to solve the linear system using Mapdl: 11.628777742385864 seconds
```

Norm of the MAPDL Solution

```mm.norm(x)
```

Out:

```1.0000000000000029
```

Solve the solution using numpy

```print(f'Solving a ({dim} x {dim}) dense linear system using numpy...')

t1 = time.time()
x_py = np.linalg.solve(a_py, b_py)
t2 = time.time()
print(f'Elapsed time to solve the linear system using numpy: {t2 - t1} seconds')
```

Out:

```Solving a (10000 x 10000) dense linear system using numpy...
Elapsed time to solve the linear system using numpy: 12.647397756576538 seconds
```

Norm of the numpy Solution

```np.linalg.norm(x_py)
```

Out:

```1.0000000000000042
```

Total running time of the script: ( 0 minutes 28.391 seconds)

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