FINITE DIFFERENCES

Finite difference method is used for complex differential equations which are difficult to solve by any methods. Finite difference is an approximation method. There are different functions or applications for finite difference method.

1. Polynomial determination for curve fitting
2. Interpolation
3. Differentiation and Integration
4. Smoothing of data

The methods for each of these applications usually introduce a formula with a certain pattern which could be used after iteration. The iteration process is taking differences between data.

POLYNOMIAL FUNCTION

The number of iteration is taken as the degree of polynomial. If there are 3 iteration steps for constant difference then the formula is:

                          

The leading coefficient being a.

INTERPOLATION FUNCTION

There are different methods for solving interpolation using finite difference with difference equation patterns.

1. Gregory-Newton Forward Interpolation Method
2. Gregory-Newton Backward Interpolation Method
3. Gregory-Newton Divided Interpolation Method
4. Lagrange Interpolation Method
5. Stirling Formula in Forward Difference
6. Gauss's Forward Interpolation Formula
7. Gauss's Backward Interpolation Formula

DIFFERENTIATION AND INTEGRATION FUNCTION

In using finite difference in solving for complex derivatives, three major methods are used:

1. Backward difference
2. Forward difference
3. Central difference

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DETERMINANTS

By making the linear equations into matrices, it would be easier to work on finding the unknowns of many equations. Although the most common way to go through the determinants is by using the basket method of the Cramer's rule, expansion of the determinants (using Laplace expansion) is also possible. This method breaks down large determinants to smaller ones.

Matrices and determinants look alike only that instead of a bracket, determinants use bars. Also, determinants have values.

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ALGEBRA OF MATRICES

Solving linear equations using simultaneous solution either by substitution or elimination is fine until 3 equations. But it gets harder to use these methods when there are more variables and equations involved.  Hence, with more of the required, matrices are used.

Matrices are sets of numbers so arranged according to the linear equation they are referred from. There are different parts as well as types of matrices used in the solution of determinants in the next topic.

credits from Matrix Fans

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