# Functions
This directory contains the functions written throught the course.
- [days.m](days.m) - computes days elapsed in a year.
- [falsePositive.m](falsePositive.m) - numberical root finding function.
- [heun.m](heun.m) - Heuns method for solving ODE's.
- [luFactor.m](luFactor.m) - LU Decomposition.
- [simpson.m](simpson.m) - numerical integration technique.
- [spacialMatrix.m](spacialMatrix.m) - function to make special matrix with a specific set of criteria.

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## [days.m](days.m)
Function to count the total days elapsed in a year according to a given date.
The syntax of the function is `days(<months>, <days>, <leap>)` where `<months>` is a integer (1-12). `<days>` is an integer from 1-31 and `<leap>` accounts for leap years.

### Input
- `months` - month number (1-12). Example: `8` represents August.
- `days` - day number of the month.
- `leap` - indicates if the year is a leapyear or a regular year. `0` for regular and `1` for leap year.

### Output
- `nd` - number of days elapsed in the year.

*Example:*
`days(8,4,0)` represents August 8th in a regular year (non-leap year).

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## [falsePositive.m](falsePositive.m)
Function finds the root of an anonymous function using the false position method.
Synopsis: `[root, fx, ea, iter] = falsePosition(func, xl, xu, es, maxit, varargin)`.

### Input
- `func` - the function being evaluated.
- `xl` - lower bound guess.
- `xu` - upper bound guess.
- `es` - desired relative error (default 0.0001%)
- `maxit` - maximum number of iterations (default 200)
- `varargin` - any additional parameters used by the function

### Output
- `root` - estimated root location.
- `fx` - function evaluated at root location.
- `ea` - approximated relative error (%).
- `iter` - number of iterations performed.

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## [heun.m](heun.m)
Uses the heun method to integrate an ODE.
Synopsis: `[t,y] = Heun(dydt,tspan,y0,h)`.

### Input
- `dydt` -the differential equation of interest (must be anonymous function).
- `tspan` - the initial and final values of the independent variable as a vector with length=2 [ti,tf].
- `y0` - the initial value of the dependent variable.
- `h` - step size.
- `es` - stopping criterion (%), optional (default = 0.001).
- `maxit` - maximum iterations of corrector, optional (default = 50).

### Output
- `t` - vector of independent variable values
- `y` - vector of solution for dependent variable

**Notes:**
This function needs some working on to compute a correct solution when using multiple steps with an irregular step size at the end.

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## [luFactor.m](luFactor.m)

Performs LU decomposition with pivoting.
Synopsis: `[L, U, P] = luFactor(A)`.

### Input
- `A` - a coefficient matrix.

### Output
- `L` - lower triangular matrix, with 1's along the diagonals.
- `U` - upper triangular matrix.
- `P` - the permutation matrix.

**Notes:**
Be cautious when using this function on bigger matrices. The `L` variable is known to be incorrect.

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## [simpson.m](simpson.m)
Evaluates the integral of two vectors by Simpsons 1/3 rule.
Synopsis: `[I] = Simpson(x, y)`

### Input
- `x` - the vector of equally spaced independent variable.
- `y` - the vector of function values with respect to x.

### Output
- `I` - numerical calculated integral.

**Notes:**
The current state of this function is *deprecated*. The algorithm fails compute the correct trapeziodal rule given 2 data points as well as 3 data points. Thus, failing to solve real problem. Pull requests are welcomed.

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## [specialMatrix.m](specialMatrix.m)
Function returns a special matrix A with the following criteria:
- The value of each element in the first row is the number of the column.
- The value of each element in the first column is the number of the row.
- The rest of the elements each has a value equal to the sum of the element above it and element to the left.
- The function returns a sensible error if the user does not input exactly two arguments.
 Synopsis: `[root, fx, ea, iter] = falsePosition(func, xl, xu, es, maxit, varargin)`.

### Input
- `input` - any additional parameters used by the function


### Output
- `A` - special matrix with the appropriate rules

**Notes:**
*Synopsis* and *input* need to be updated. This function has not much of a practical application, rather a very good exercise for beginners to get started with the basics of matrix manipulation and user-defined functions.