Grade 12 STEM Mathematics: Numerical Methods

Flashcards covering the essential vocabulary, error analysis types, linear algebra concepts, interpolation methods, numerical integration, and differential equation solutions from Grade 12 STEM Mathematics.

Numerical Analysis

A discipline of mathematics that analyses the methods and algorithms used to solve a wide range of problems including mathematics, physics, engineering, medicine, and the social and life sciences.

Scientific Notation

A notation where a real number $k$ is written in the form $k = m \times 10^n$, where $1 \le m \le 10$ and the exponent $n$ is an integer.

Numerical Error

The difference between the exact solution $x$ and its approximate solution $x^*$, expressed as $e = x - x^*$.

Blunders (Gross Errors)

Human errors caused by mistakes and oversight during scientific investigations.

Modelling Errors

Also known as formulation errors, these arise during the modelling process when factors are ignored to simplify the problem.

Data Uncertainty

Also known as data errors, these are due to the uncertainty of the physical problem data.

Discretization Errors

Errors resulting from approximating and replacing complex continuous problems with discrete representations.

Absolute Error

The absolute value of the error $e$, defined as $\hat{e} = |x - x^*|$.

Relative Error

The ratio between the absolute error $\hat{e}$ and the absolute value of the exact solution $x$, defined as $\tilde{e} = \frac{|x - x^*|}{|x|}$ for $x \neq 0$.

Round-off Error

The error caused by replacing a number by its closest machine number due to computers representing numbers with a finite number of digits.

Truncation Errors (Chopping Errors)

Errors occur when chopping an infinite number and replacing it by a finite number, or by truncating a series after a finite number of terms.

Taylor Series

An infinite series representation of an infinitely-differentiable function $f(x)$ at point $a$, defined as $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$.

Maclaurin Series

A Taylor series expansion specifically for the case where the point of expansion $a = 0$.

Numerical Stability

A property of a numerical algorithm where the final result is relatively unaffected by perturbations during the computation process.

Conditioning

Also called sensitivity, it is a measure of how sensitive the output is to small changes in the input data.

Condition Number of a Function

If $f(x)$ is differentiable at $x$, the condition number is $Cond(f(x)) = \frac{|x f'(x)|}{|f(x)|}$ for $f(x) \ne 0$.

Accuracy

A measure of the closeness of the approximate solution to the exact solution.

Precision

A measure of the closeness of two or more measurements to each other.

Matrix

A rectangular array of numbers or functions enclosed in brackets; the items are called elements.

Square Matrix

A matrix that has an equal number of rows and columns.

Main Diagonal

The diagonal in a square matrix containing the entries $a_{11}, a_{22}, \dots, a_{nn}$.

Vector

A matrix having just a single row (row vector) or a single column (column vector).

Scalar Multiplication

The product of an $m \times n$ matrix $A = [a_{jk}]$ and a scalar $c$, resulting in a matrix where every entry is multiplied by $c$.

Zero Matrix

A matrix of size $m \times n$ where all elements are zero.

Identity Matrix

A diagonal matrix, denoted by $I$, whose diagonal elements are all $1\text{s}$, satisfying $IA = AI = A$ for any square matrix $A$.

Inverse Matrix

For a given matrix $A$, a matrix $B$ (denoted $A^{-1}$) such that $AB = BA = I$.

Singular Matrix

A matrix that does not have an inverse, also referred to as non-invertible; this occurs when the determinant is zero.

Augmented Matrix

A matrix representing a system of equations containing the coefficient entries and the constants separated by a vertical line.

Minor ($M_{ij}$)

The determinant of the submatrix obtained by deleting the $i\text{-th}$ row and $j\text{-th}$ column of a matrix.

Co-factor ($A_{ij}$)

The value associated with a minor, defined as $A_{ij} = (-1)^{i+j} M_{ij}$.

Gaussian Elimination

A direct method for solving systems of equations by using legal row operations to convert the coefficient matrix into an upper triangular matrix.

Gauss-Jordan Method

A direct method that converts the coefficient matrix into an identity matrix using legal row operations.

Jacobi Iterative Method

A technique for solving linear systems where the components for the current iteration are generated solely from the components of the previous iteration.

Gauss-Seidel Iterative Method

An iterative technique where the most recently calculated components of the current iteration are used to compute subsequent components.

Successive Over-Relaxation (SOR) Method

An iterative method particularly useful for solving linear systems occurring in numerical solutions of partial-differential equations, often using a factor $w > 1$.

Interpolation

The focus of providing a credible projection or fitting a function to given data, such as population census figures.

Lagrange Interpolating Polynomial

The unique polynomial of degree at most $n$ that agrees with a function $f$ at $n+1$ distinct numbers.

Neville's Method

A recursive process used to find the $n\text{th}$ Lagrange interpolating polynomial for a set of points.

Divided Difference

A recursive formula (e.g., $f[x_i, x_{i+1}]$) used in Newton's interpolation methods to represent function values across distinct nodes.

Hermite Interpolation

The unique polynomial of least degree that agrees with both function values $f$ and their derivatives $f'$ at a given set of nodes.

Cubic Spline Interpolation

A piecewise-polynomial approximation employing cubic polynomials between succeeding pairs of nodes to ensure continuous differentiability and continuous second derivatives.

Midpoint Rule

A numerical integration technique that estimates an integral by summing the values of the function evaluated at the midpoints of sub-intervals.

Trapezoidal Rule

A numerical integration method that approximates the value of a definite integral by dividing the interval into sub-intervals and treating the segments as trapezoids.

Simpson's Rule

A numerical integration technique that uses piecewise quadratic functions to approximate the curve of a function.

Gaussian Quadrature

A method to approximate the integral of $f(x)$ between $-1$ and $1$ using the sum $\sum_{i=1}^{n} c_i f(x_i)$ that is exact for polynomials of degree at most $2n - 1$.

Euler's Method

A first-order numerical technique for solving initial value problems based on the Taylor's Series.

Heun's Method

A numerical method for solving initial value problems that provides an improved approximation over the simple Euler's method.

Runge-Kutta Order Four Method

A highly popular and accurate numerical method used to solve initial value problems for ordinary differential equations.

Eigenvector

A nonzero vector that changes at most by a scalar factor (the eigenvalue) when a linear transformation is applied to it.

Eigenvalue (\lambda)

The scalar factor by which an eigenvector is scaled during a linear transformation.

Characteristic Polynomial

The polynomial defined as $p(\lambda) = \det(A - \lambda I)$, used to find eigenvalues.

Characteristic Equation

The equation obtained by setting the characteristic polynomial to zero: $p(\lambda) = 0$.