# Math 410: Intro to Numerical Methods

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## Catalog Information

### Title

Introduction to Numerical Methods.

(3:3:0)

F

### Description

Root finding, interpolation, curve fitting, numerical differentiation and integration, multiple integrals, direct solvers for linear systems, least squares, rational approximations, Fourier and other orthogonal methods. [This official course description appears to differ with current standard practice, in that iterative solvers of linear systems are taught in this course, while "Fourier and other orthogonal methods" are postponed until Math 411.]

## Desired Learning Outcomes

### Prerequisites

Students are required to have had multivariable calculus.

### Minimal learning outcomes

Students should be able to describe, derive, and implement the numerical methods listed below. They should be able to explain the advantages and disadvantages of each method. They should understand error analysis and be able to make practical decisions based on the outcomes of that analysis.

1. Numerical solution of equations of one variable
• Bisection method
• Secant method
• Fixed-point iteration
• Newton's method
• Error analysis
• Polynomial equations
2. Interpolation
• Lagrange interpolation
• Divided-difference methods
• Hermite interpolation
• Cubic spline interpolation
3. Numerical differentiation
• Derivation of formulas
• Backward-difference
• Forward-difference
• Centered-difference
• Error analysis
• Richardson's extrapolation
4. Numerical integration
• Newton-Cotes formulas
• Composite integration
• Adaptive quadrature
• Gaussian quadrature
• Multiple integrals
• Error analysis
5. Numerical solution of linear systems
• Direct methods
• Gaussian elimination
• Pivoting strategies
• Factorization methods
• Iterative methods
• Jacobi iteration
• Gauss-Seidel iteration
• Relaxation methods

### Courses for which this course is prerequisite

Math 410 is the introductory numerical analysis course and is a prerequisite for the other 3 numerical analysis courses: Math 411, 510, and 511. It is also a prerequisite for Math 480.