Math Ladder · College

Numerical Analysis

How computers do calculus and linear algebra: floating-point error, root finding, solving systems, interpolation, numerical integration, ODE solvers and eigenvalues.

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45 topics in 7 units. Builds on Calculus 2: Integrals and Series and Calculus 3: Multivariable Calculus and Linear Algebra and Differential Equations. Not sure this is the right course? Find your rung.

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Unit 1 Numbers on a Computer

8 topics

See how a computer stores numbers, where errors come from, and how to tell a hard problem from a bad method.

  1. Absolute and Relative Error
  2. Binary Fractions
  3. Floating-Point Numbers
  4. Round-Off and Cancellation
  5. Truncation Error and Order of Accuracy
  6. Conditioning of a Problem
  7. Stable and Unstable Algorithms
  8. Horner's Method

Unit 2 Solving Equations

6 topics

Find roots with bisection, fixed-point iteration, Newton's method and the secant method, and measure how fast each one converges.

  1. The Bisection Method
  2. Fixed-Point Iteration
  3. Order of Convergence
  4. Newton's Method: Speed and Failure
  5. The Secant Method
  6. Newton's Method for Systems

Unit 3 Linear Systems on a Computer

7 topics

Solve Ax = b reliably: the cost of elimination, pivoting, LU, norms and condition numbers, and iterative methods.

  1. The Cost of Elimination
  2. Partial Pivoting
  3. Solving with PA = LU
  4. Vector and Matrix Norms
  5. The Condition Number of a Matrix
  6. Jacobi Iteration
  7. Gauss–Seidel Iteration

Unit 4 Interpolation

5 topics

Pass polynomials and splines through data points, and see when adding points helps and when it hurts.

  1. Lagrange Interpolation
  2. Newton's Divided Differences
  3. Interpolation Error
  4. Runge's Phenomenon and Chebyshev Nodes
  5. Cubic Splines

Unit 5 Derivatives and Integrals

7 topics

Approximate derivatives and integrals, choose step sizes, and squeeze out extra accuracy with extrapolation and smart nodes.

  1. Finite-Difference Derivatives
  2. The Best Step Size
  3. Richardson Extrapolation
  4. Error in the Trapezoid and Simpson's Rules
  5. Romberg Integration
  6. Gaussian Quadrature
  7. Adaptive Integration

Unit 6 Differential Equations

7 topics

Measure the error of ODE solvers, solve systems, keep steps stable, and turn boundary value problems and the heat equation into linear algebra.

  1. Local and Global Error
  2. Systems and Higher-Order Equations
  3. Stability of Euler's Method
  4. Backward Euler and Stiff Equations
  5. Multistep Methods
  6. Boundary Value Problems by Finite Differences
  7. The Heat Equation: An Explicit Scheme

Unit 7 Least Squares and Eigenvalues

5 topics

Fit data stably with QR, and find eigenvalues by iteration.

  1. Least Squares with QR
  2. The Power Method
  3. Inverse and Shifted Power Methods
  4. Gershgorin Disks
  5. The QR Algorithm
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