Numerical Methods

Numerical Methods involves algorithms and techniques for solving mathematical problems numerically rather than analytically.

Introduction to Numerical MethodsSolving Nonlinear EquationsNumerical Differentiation and Integration
Solving Systems of Linear EquationsNumerical Solutions of Differential EquationsInterpolation and Curve Fitting
Engineering SimulationsMedical Imaging and DiagnosticsEnvironmental Modeling
Breaking Problems Into StepsUsing Technology Wisely
Introduction to Nume...Solving Nonlinear Eq...Numerical Differenti...Solving Systems of L...Numerical Solutions ...Interpolation and Cu...Engineering Simulati...Medical Imaging and ...Environmental Modeli...Breaking Problems In...Using Technology Wis...
Advanced Topics

Solving Systems of Linear Equations

Tackling Multiple Equations at Once

Many problems involve not just one, but several equations and unknowns. Numerical methods let us solve these systems efficiently.

Gaussian Elimination

  • Arrange the equations in a matrix.
  • Use row operations to reduce the matrix to a simpler form.
  • Back-substitute to find the unknowns.

Iterative Methods

  • Jacobi Method: Updates each variable using old values from the previous step.
  • Gauss-Seidel Method: Uses the newest values as soon as they are available.

When Are These Used?

  • When the system is too big for hand calculation.
  • When the equations can't be solved exactly.

Applications

These methods are essential in engineering, computer graphics, physics, and economics.

Key Formula

\[Ax = b\]

Examples

  • Calculating currents in a complex electrical circuit.

  • Finding the forces in a bridge's support beams.

In a Nutshell

Efficient techniques to solve lots of equations at once, even when computers are needed.

Key Terms

Matrix
A rectangular array of numbers.
Iterative Method
A technique that approaches a solution through repeated steps.
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Numerical Solutions of Differential Equations