Numerical Analysis     
          
1.  Approximations and Errors
       1.1  Taylor series approximations  
                 1.1.1  Taylor series and their error
                 1.1.2  Calculations with Mathematica
       1.2  Absolute and relative errors
       1.3  Propogation of errors - the function rule
                 1.3.1  The function rule
                 1.3.2  Calculations with Mathematica
       1.4  The algebra of errors
       1.5  Floating point numbers and round-off errors
                 1.5.1  Floating point numbers
                 1.5.2  Floating point number computations using Mathematica
       1.6  Round-off errors in floating point computations
                 1.6.1  Round-off errors
                 1.6.2  Propogation of errors in floating point computations using Mathematica
       1.7  Other ways to specify errors
                 1.7.1  Number of significant digits and intervals
                 1.7.2  Representing errors in Mathematica
                 1.7.3  Error calculations based on interval computations
       1.8  Binary floating point numbers

2.  Nonlinear Equations
       2.1  General principles of solving nonlinear equations
                 2.1.1 - 2.1.6  Existence, uniqueness and accuracy of an approximation
                 2.1.7          Computations with Mathematica
                 2.1.8	        Computations with MATLAB
       2.2  Bisection method
       2.3  Newtons method
       2.4  Secant method
       2.5  Fixed point iteration
       2.6  Equations involving parameters

3.  Systems of Equations
  3.1	Systems of equations
                3.1.1	Systems of equations
                3.1.2	Solving Linear Systems with Mathematica
                3.1.3	Solving linear equations with MATLAB
  3.2	LU decompositions without pivoting
                3.2.1	The alogorithm
                3.2.2	The LU decomposition algorithm in Mathematica
  3.3	LU decompositions with pivoting
                3.3.1	The alogorithm
                3.3.3   The LU decompositions with pivoting in Mathematica
                3.3.3	Mathematica's LU decomposition function
                3.3.4	LU decompostions with MATLAB
                3.3.5	The time it takes to solve linear equations
  3.4	Norms and condition numbers
                3.4.1	Norms of Vectors
                3.4.2	Norms of Matrices
                3.4.4	Propagation of errors in linear formulas – componentwise analysis
                3.4.5	Norms and condition numbers
                3.4.6	Norms in MATLAB
  3.5	Systems of nonlinear equations

4.  Approximation
   4.1	Polynomial Interpolation
           4.1.1	Polynomial interpolation in Mathematica
           4.1.2	Polynomial interpolation in MATLAB
   4.2	Divided Differences 
           4.2.1	Divided differences in Mathematica
  4.3	The error in the interpolating polynomial
           4.3.1	Approximating mathematical functions with polynomials
           4.3.2	The error in the interpolating polynomial due to errors in the data
  4.4	Piecewise polynomial interpolation and splines
           4.4.1	Piecewise polnomial interpolation in Mathematica
           4.4.2	Cubic splines
  4.5	Least Squares Curve Fitting

5.  Differentiation and Integration
  5.1	Approximating derivatives
  5.2	The trapezoidal rule
  5.3	Simpson's rule
  5.4	The Midpoint Rule
  5.5	Construction of quadrature rules
  5.6	Error formulas for quadrature rules
  5.7	Gaussian quadrature

6.	Differentiation and Integration
  6.1	Introduction
  6.2	Euler's method
  6.3	The trapezoidal method
  6.4	The midpoint method
  6.5	Runge-Kutta methods
           6.5.2	The classical Runge-Kutta method
           6.5.3	Three & four stage Runge-Kutta methods
  6.6	Systems of differential equations
  6.7	Second order differential equations

7	Bibliography


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