# APDL Math Overview¶

APDL Math provides the ability to access and manipulate the large sparse matrices and solve a variety of eigenproblems. PyMAPDL classes and bindings present APDL Math in a similar manner to the popular numpy and scipy libraries. The APDL Math command set is based on tools for manipulating large mathematical matrices and vectors that provide access to standard linear algebra operations, access to the powerful sparse linear solvers of ANSYS Mechanical APDL (MAPDL), and the ability to solve eigenproblems.

Python and MATLAB’s eigensolver is based on the publicly available LAPACK libraries and provides reasonable solve time for relatively small DOF (degree of freedom) eigenproblems of perhaps 100,000. However, ANSYS’s solvers are designed for the scale of 100s of millions of DOF and there are a variety of situations where users can directly leverage ANSYS’s high performance solvers on a variety of eigenproblems. Fortunately, you can leverage this without relearning an entirely new language as this has been written in a similar manner as `numpy` and `scipy`. For example, here is a comparison between the `scipy` linear algebra solver and ANSYS’s solver:

`numpy` vs PyMAPDL Math Implementation

`numpy` and `scipy`

`ansys.mapdl.math`

```k_py = k + sparse.triu(k, 1).T
m_py = m + sparse.triu(m, 1).T
n = 10
ev = linalg.eigsh(k_py, k=neqv, M=m_py)
```
```k = mm.matrix(k_py, triu=True)
m = mm.matrix(m_py, triu=True)
n = 10
ev = mm.eigs(n, k, m)
```

What follows is a basic example and a detailed description of the PyMAPDL Math API. For additional PyMAPDL Math examples, visit PyMapdl Math Examples.

## MAPDL Matrix Example¶

This example demonstrates how to send an MAPDL Math matrix from MAPDL to Python and then send it back to be solved. While this example runs `MapdlMath.eigs()` on mass and stiffness matrices generated from MAPDL, you could instead use mass and stiffness matrices generated from an external FEM tool, or even modify the mass and stiffness matrices within Python.

First, solve the first 10 modes of a 1 x 1 x 1 steel meter cube in MAPDL.

```import re

from ansys.mapdl.core import launch_mapdl
mapdl = launch_mapdl()

# setup the full file
mapdl.prep7()
mapdl.block(0, 1, 0, 1, 0, 1)
mapdl.et(1, 186)
mapdl.esize(0.5)
mapdl.vmesh('all')

# Define a material (nominal steel in SI)
mapdl.mp('EX', 1, 210E9)  # Elastic moduli in Pa (kg/(m*s**2))
mapdl.mp('DENS', 1, 7800)  # Density in kg/m3
mapdl.mp('NUXY', 1, 0.3)  # Poisson's Ratio

# solve first 10 non-trivial modes
out = mapdl.modal_analysis(nmode=10, freqb=1)

# store the first 10 natural frequencies
mapdl.post1()
resp = mapdl.set('LIST')
w_n = np.array(re.findall(r'\s\d*\.\d\s', resp), np.float32)
print(w_n)
```

We now have solved for the first 10 modes of the cube:

```[1475.1 1475.1 2018.8 2018.8 2018.8 2024.8 2024.8 2024.8 2242.2 2274.8]
```

Next, load the mass and stiffness matrices that are stored by default at `<jobname>.full`. First, create an instance of `MapdlMath` as `mm`:

```mm = mapdl.math

# load by default from file.full
k = mm.stiff()
m = mm.mass()

# convert to numpy
k_py = k.asarray()
m_py = m.asarray()
mapdl.clear()
print(k_py)
```

These matrices are now solely stored within Python now that we’ve run `Mapdl.clear()`.

```(0, 0)      37019230769.223404
(0, 1)      10283119658.117708
(0, 2)      10283119658.117706
:   :
(240, 241)  11217948717.943113
(241, 241)  50854700854.68495
(242, 242)  95726495726.47179
```

The final step is to send these matrices back to MAPDL to be solved. While we have cleared MAPDL, we could have shutdown MAPDL, or even transferred them to a different MAPDL session to be solved.

```my_stiff = mm.matrix(k_py, triu=True)
my_mass = mm.matrix(m_py, triu=True)

# solve for the first 10 modes above 1 Hz
nmode = 10
mapdl_vec = mm.eigs(nmode, my_stiff, my_mass, fmin=1)
eigval = mapdl_vec.asarray()
print(eigval)
```

As expected, the natural frequencies obtained from `MapdlMath.eigs()` is identical to the result from `Mapdl.solve()` within MAPDL.

```[1475.1333421  1475.1333426  2018.83737064 2018.83737109 2018.83737237
2024.78684466 2024.78684561 2024.7868466  2242.21532585 2274.82997741]
```

If you wish to obtain the eigenvectors as well as the eigenvalues, initialize a matrix `eigvec` and send that to `MapdlMath.eigs()`:

```nmode = 10
eigvec = mm.zeros(my_stiff.nrow, nmode)  # for eigenvectors
val = mm.eigs(nmode, my_stiff, my_mass, fmin=1)
```

The MAPDL Math matrix `eigvec` now contains the eigenvectors for the solution.