# SP4E - Homework 1

## General info

This file provides a brief documentation and information related to the first Homework of the course Scientific Programming for Engineers, fall 2019.

This homework is done by **O. Ashtari** and **A. Sieber**

Last update: 16.10.2019

## Project Description
This project is intended to solve a minimization problem on n-dimensional quadratic functions defined as S(X)= XᵀAX - Xᵀb where Xᵀ =[x1, x2, ..., xn], T represents the transpose operation, A is an n by n square matrix and b a column vector of size n.
	
## Requirements
The project is created with Python 3.7. It moreover requires the following libraries to work properly:
* numpy
* argparse
* scipy
* matplotlib
* sys
* math
	
## How to Use

### Program 1: ***optimizer.py***

This script solves minimization problems using the scipy.optimize.minimize routine. The minimization solver can be specified as an input argumeent by entering the following command line:

```
$ python3 optimizer.py method
```
Where the variable  method is one of the following:

* Nelder-Mead
* Powell
* CG (default)
* BFGS
* L-BFGS-B
* TNC
* SLSQP

The initial guess of the minimization process has to be specified directly in the optimizer.py file by changing the value of the variable X0. The matrix A and vector B can also be modified directly in the file.

### conjugate_gradient.py

### post_processing.py

Post-processing file that takes as inputs the value of A, B as well as the intermediate solutions of the iterative minimization process and the method used. The file generates a 3D plot displaying the function to be minimized as well as the intermediate solutions.

### Program 2: conjugate_gradient.py
Based on the problem description in the homework sheet, the objective function here is:

S(X)= ½(XᵀAX) - Xᵀb

Therefore, the user should note the factor 1/2 here which means definition of matrix A here is different from that in Program 1.

To run the program, the objective function should be defined by getting matrix A and vector b, and the code should be provided with an starting point x₀ as well. These three are mandatory arguments, followed after `-A`, `-b`, and `-x0` which should be entered in rowmajor format separated by space. For example the command below runs the program for:
A=[ [1, 2] , [3, 4] ]   ,   bᵀ=[5,6]   ,   x₀ᵀ=[7,8]

```
$  python3  conjugate_gradient.py  -A 1 2 3 4  -b 5 6  -x0 7 8
```