Week 8: Solving Inverse Problems

This set of flashcards covers key vocabulary and concepts in numerical methods and inverse problems for the COMP1006 Engineering Programming module.

Numerical Methods

Algorithms for approximating solutions to mathematical problems difficult or impossible to solve analytically.

Inverse Problems

Problems where desired outputs are known, and the goal is to estimate inputs or construct a model that produces those outputs.

Bisection Method

Iteration method based on the Intermediate Value Theorem for finding roots.

Newton-Raphson Method

A numerical method that uses the rate of change of a function to find roots more efficiently than the bisection method.

Secant Method

An iterative method that approximates the derivative of a function to find roots when the exact derivative is unavailable.

Intermediate Value Theorem

A continuous function over an interval takes on all values between its endpoints.

Convergence Rate

The speed at which an iterative method approaches the solution.

Linear Convergence Rate

Describes methods like the bisection method, where the error decreases linearly with iterations.

Quadratic Convergence Rate

Describes methods like Newton-Raphson where error decreases quadratically, resulting in rapid convergence.

Superlinear Convergence Rate

A convergence rate of the secant method, which is faster than linear but slower than quadratic.

Forward Finite Difference

An approximation for the derivative that uses the function value at a point and at a point slightly ahead.

Backward Finite Difference

An approximation for the derivative that uses the function value at a point and at a point slightly behind.

Scalar Variable

A single value, as opposed to a vector or matrix.

Closed-form Solution

An explicit expression or solution to a mathematical problem.

CORDIC Algorithm

An algorithm used for calculating trigonometric functions and other mathematical operations.

Initial Guesses

Starting points necessary for iterative methods to converge to the correct root.

Optimization Problems

Problems where an optimal solution is sought from given constraints. Unconstrained optimization problems do not have constraints on the variables.

Power Series Representation

Expressing functions as an infinite sum of terms calculated from the values of their derivatives.

Root Finding

The process of determining the values of x for which a given function evaluates to zero.

Taylor Approximation

An approximation of a function as a polynomial derived from its derivatives at a specified point.

Hybrid Root Finder

A method that combines different root-finding techniques for improved performance.