Numerical Methods | Index

Numerical methods are step-by-step procedures for finding approximate answers to problems that are hard or impossible to solve exactly, along with ways to measure how far off the answers can be.

The course covers error and computer arithmetic, root finding, interpolation, numerical differentiation and integration, numerical linear algebra and numerical solutions of differential equations.

It’s for college engineering, math and computer science students, usually in the second or third year.

What to know first: Calculus 2, especially Taylor polynomials, plus first-order equations in Differential Equations, Unit 1. Linear Algebra and partial derivatives from Calculus 3 are needed for Unit 5.

Lessons for this course are coming soon. The list below shows every planned lesson in teaching order.

Units in this course

Unit 1: Error and Computer Arithmetic

  1. Absolute and Relative Error (coming soon) — measure how far an approximation is from the true value
  2. Floating-Point Arithmetic (coming soon) — how computers store numbers and where round-off error comes from
  3. Loss of Significance (coming soon) — avoid subtracting nearly equal numbers by rewriting formulas

Also see: Taylor polynomials and the Lagrange error bound (Calculus 2).

Unit 2: Root Finding

  1. Bisection Method (coming soon) — trap a root by halving an interval where the function changes sign
  2. Fixed-Point Iteration (coming soon) — solve x = g(x) by repeating g and tell when it converges
  3. Secant Method (coming soon) — find a root with secant lines instead of derivatives
  4. False Position Method (coming soon) — combine the bisection and secant ideas to find a root
  5. Order of Convergence (coming soon) — measure how fast an iterative method closes in on a root, from linear to quadratic

Also see: Newton’s method and the intermediate value theorem (Calculus 1).

Unit 3: Interpolation

  1. Lagrange Interpolation (coming soon) — find the polynomial that passes through a set of points
  2. Newton’s Divided Differences (coming soon) — build an interpolating polynomial one point at a time
  3. Interpolation Error (coming soon) — bound the error of an interpolating polynomial and see Runge’s phenomenon
  4. Cubic Splines (coming soon) — join points with smooth piecewise cubic curves

Unit 4: Numerical Differentiation and Integration

  1. Numerical Differentiation (coming soon) — estimate derivatives with forward, backward and central differences
  2. Richardson Extrapolation (coming soon) — combine two estimates to cancel the leading error term
  3. Romberg Integration (coming soon) — improve trapezoidal rule estimates with Richardson extrapolation
  4. Gaussian Quadrature (coming soon) — choose nodes and weights for the most accurate integral estimate

Also see: Riemann sums and the trapezoidal rule (Calculus 1), and Simpson’s rule (Calculus 2).

Unit 5: Numerical Linear Algebra and Nonlinear Systems

  1. Partial Pivoting (coming soon) — use Gaussian elimination with row swaps to control round-off error
  2. Jacobi Method (coming soon) — solve a linear system by iterating on each variable
  3. Gauss-Seidel Method (coming soon) — use each new value right away, which usually converges faster than the Jacobi method
  4. Condition Number of a Matrix (coming soon) — measure how sensitive a linear system is to small errors
  5. Power Method (coming soon) — approximate the eigenvalue of largest absolute value (the dominant eigenvalue) and its eigenvector
  6. Newton’s Method for Systems (coming soon) — solve systems of nonlinear equations with the Jacobian

Also see: Gaussian elimination, LU decomposition and eigenvalues in Linear Algebra.

Unit 6: Numerical Solutions of Differential Equations

  1. Heun’s Method (coming soon) — improve Euler’s method with a predictor-corrector step
  2. Multistep Methods (coming soon) — use several earlier points to take each step with Adams-Bashforth and Adams-Moulton methods
  3. Stiff Differential Equations (coming soon) — recognize stiff equations and solve them with the backward Euler method
  4. Shooting Method (coming soon) — solve a boundary value problem by adjusting an initial-value problem
  5. Finite Difference Method for Boundary Value Problems (coming soon) — solve boundary value problems by replacing derivatives with differences

Also see: Euler’s method (Calculus 2) and the Runge-Kutta method in Differential Equations, Unit 1.

For Teachers: Standards Alignment

Lessons follow a typical undergraduate course in numerical methods (numerical analysis) for engineering, math and computer science majors. There is no AP exam for this course. Methods that first appear in calculus live in those courses: Newton’s method and the trapezoidal rule in Calculus 1, Simpson’s rule and Euler’s method in Calculus 2, and the Runge-Kutta method in Differential Equations.

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