# Convergence Behavior of Several Solvers in GAMS and Matlab

In this post I wanted to document the behavior of several different numerical methods using a set of benchmark test problems.

### Test Problems

- Ball sliding on plane
- Block sliding on plane
- Block sliding from rest on incline
- Block coming to resting on incline
- Ball falling on stack of 20 balls

### Solvers

Solver (Matlab) | Solver (Gams) | |
---|---|---|

Nesterov | Conopt | |

Barzilai-Borwein | IPopt | |

Jacobi | Path |

## Test Problems

### Ball sliding on plane

In this test problem there is a ball in perfect contact with a plane, the ball has an initial velocity of in the direction. friction is .1, step size = .005, with a step size of .01, path returns a weird solution.

The next plots shows the rotational velocity in the z direction

### Box sliding on plane

In this test problem there is a box in perfect contact with a plane, the box has an initial velocity of in the direction. The friction coefficient is .3;s

### Box sliding on Incline

In this test problem there is a box in perfect contact with a plane, it starts at rests and then slides down for one second. friction is .2

### Box sliding to a stop on Incline

In this test problem there is a box in perfect contact with a plane, it starts with velocity of parallel to the incline. It sides and comes to a stop, friction is .6

### Stack of 20 balls with heavy mass

A stack of 20 spheres with a larger heavier sphere dropping on top

Velocity of the top ball: