Polynomial Curve Fitting

EXAMPLES :

 



 

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Homogeneous System

A homogeneous system of linear algebraic equations is one in which all the numbers on the right hand side are equal to 0:

a11x1 + : : : + a1nxn = 0

...

am1x1 + : : : + amnxn = 0

EXAMPLE:

 

(0,0,0) - trivial or obvious solution

Gauss- Jordan Elimination

• Gauss- Jordan Elimination

                    - A method of solving a linear system of equations. This is done by transforming the system's augmented matrix into reduced row-echelon form by means of row operations.

1. Write the Augmented Matrix
2. Target: Reduced Row Echelon Form
3. Rewrite in linear systems

Gaussian Elemination with Back Substitution

1. Write the augmented matrix
2. Use Elementary Row Operations to obtain Row Echelon form of the matrix
3. Rewrite in linear system
4.Back Substitution

Row-Echelon Form

 

In linear algebra, a matrix is in echelon form if it has the shape resulting of a Gaussian elimination. Row echelon form means that Gaussian elimination has operated on the rows and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices.

 

Specifically, a matrix is in row echelon form if

 

Example:

 A 3×5 matrix in row echelon form:

Elementary Row Operations

Three operations the can be used on a system of linear equations to produce equivalent systems.

1. Interchange two equations.
2. Multiply an equation by a nonzero constant.
3. Add a multiple of an equation to another equation.

Two types of Matrix

• Augmented Matrix - the matrix derived from the coefficients and constant terms of a system of linear equations.

Example:

• Coefficient Matrix - the matrix containing only the coefficients of the system.

Example: