Grade 12 STEM Mathematics: Numerical Methods

Numerical Analysis and Error Definitions

• Numerical Analysis: A mathematical discipline focusing on algorithms for approximating solutions across sciences and engineering. Key areas include stability, convergence, accuracy, consistency, and error analysis.

• Ultima Thulë Ltd: The entity engaged by the Department of Education of Papua New Guinea to develop the STEM Curriculum.

• Scientific Notation: Defined as $k = m \times 10^n$, where 1 \le m < 10 and $n \in \mathbb{Z}$.

• Numerical Error ($e$): The difference between the exact solution ($x$) and approximate solution ($x^*$):     $e = x - x^*$

• Error Types:     * Absolute Error ($\hat{e}$): $\hat{e} = |x - x^*|$     * Relative Error ($\tilde{e}$): $\tilde{e} = \frac{|x - x^*|}{|x|}$, for $x \neq 0$.

• Sources of Error: Includes human blunders (gross errors), modeling/formulation errors, data uncertainty, and discretization errors.

• Round-Off and Truncation: Round-off error occurs from finite digit representation; truncation (or chopping) occurs by ignoring digits after a certain point or terminating an infinite series.

Polynomial Expansion and Stability

• Taylor Series: Used for function approximation near a point $a$:     $f(x) = f(a) + \frac{f'(a)}{1!} (x - a) + \frac{f''(a)}{2!} (x - a)^2 + \dots + R_n(x)$

• Maclaurin Series: A Taylor series centered at $a = 0$.

• Numerical Stability: A process is stable if small perturbations during computation do not significantly affect the final result.

• Conditioning: Measures output sensitivity to input changes.     * Condition Number of a Function: $\text{Cond}(f(x)) = \left| \frac{x f'(x)}{f(x)} \right|$     * Condition Number of a Matrix: $\text{Cond}(A) = \|A\| \|A^{-1}\|$     * Interpretation: A problem is ill-conditioned (sensitive) if $\text{Cond} \ge 1$ and well-conditioned if \text{Cond} < 1.

• Accuracy vs. Precision: Accuracy is proximity to the exact value; precision is the closeness of multiple measurements to each other.

Matrices and Linear Systems

• Matrix Definitions: A rectangular array of elements. Types include row vectors ($1 \times n$), column vectors ($m \times 1$), and square matrices ($m = n$).

• Special Matrices:     * Identity ($I$): Diagonal elements are $1$, all others are $0$.     * Inverse ($A^{-1}$): Satisfies $A A^{-1} = A^{-1} A = I$.     * **Determinant (\text{det}(A)$)**: For a 2 \times 2$matrix$[a, b; c, d]$, it is$ad - bc$. If$\text{det}(A) = 0, the matrix is singular (non-invertible).\n* **Augmented Matrix**: Represents a system of linear equations by combining coefficients and constants ([A|b]).\n* **Legal Row Operations**: \n    1. Multiply a row by a constant (cR_i).\n    2. Interchange two rows (R_i \leftrightarrow R_j).\n    3. Add a constant multiple of one row to another (R_i + cR_j).\n\n# Solution Techniques for Linear Systems\n\n* **Gaussian Elimination**: Converting the coefficient matrix to upper triangular form followed by backward substitution.\n* **Gauss-Jordan Method**: Converting the coefficient matrix into an Identity matrix.\n* **Cramer’s Rule**: Uses determinants to solve for variables: x_i = \frac{\text{det}(A_i)}{\text{det}(A)}$, where$A_i$is the matrix formed by replacing the$i-th column with the constant vector.\n* **Iterative Methods**:\n    * **Jacobi Method**: Updates components based only on values from the previous iteration: x_i^{(k)} = \frac{1}{a_{ii}} [b_i - \sum_{j \neq i} a_{ij} x_j^{(k-1)}].\n    * **Gauss-Seidel Method**: Updates components using the most recently calculated values within the same iteration.\n    * **Successive Over-Relaxation (SOR)**: Incorporates a relaxation factor (w > 1$) to accelerate convergence:$x_i^{(k)} = (1-w)x_i^{(k-1)} + w[ \dots ].\n\n# Interpolation and Approximation\n\n* **Lagrange Interpolating Polynomial**: P(x) = \sum_{k=0}^n f(x_k) L_{n,k}(x)$, where$L_{n,k}(x) = \prod_{i=0, i \neq k}^n \frac{x - x_i}{x_k - x_i}.\n* **Neville’s Method**: A recursive approach to evaluating interpolating polynomials.\n* **Newton’s Divided Difference**: Forward and backward formulas using tabular differences to construct polynomials.\n* **Hermite Interpolation**: Constructing a polynomial that matches both function values and their derivatives at given nodes.\n* **Cubic Spline Interpolation**: Piecewise-polynomial approximation using cubic polynomials between nodes. \n    * **Natural Spline**: Continuous second derivatives with S''(x_0) = S''(x_n) = 0.\n    * **Clamped Spline**: First derivatives are specified at endpoints.\n\n# Numerical Integration\n\n* **Midpoint Rule**: M_n = \sum f(m_i) \Delta x$, where$m_i is the sub-interval midpoint.\n* **Trapezoidal Rule**: Integrates using trapezoids: T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + \dots + f(x_n)].\n* **Simpson’s Rule**: Uses piecewise quadratic approximations: S_n = \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + \dots + f(x_n)]$. Requires an even number of sub-intervals ($n).\n* **Gaussian Quadrature**: Approximates integrals using weights (c_i$) and nodes ($x_i$):$\int_{-1}^1 f(x) \,dx \approx \sum c_i f(x_i).\n\n# Numerical Solutions to ODEs\n\n* **Initial Value Problem (IVP)**: Solving \frac{dy}{dx} = f(x, y)$given$y(a) = y_0.\n* **Euler’s Method**: y_{i+1} = y_i + h f(x_i, y_i).\n* **Heun’s Method**: An improved Euler method using an average of slopes at the start and end of the interval.\n* **Runge-Kutta Method (Order Four)**: Uses four weighted slopes (K_1, K_2, K_3, K_4) to calculate the next point with high accuracy.\n\n# Eigenvalues and Eigenvectors\n\n* **Characteristic Polynomial**: p(\lambda) = \text{det}(A - \lambda I).\n* **Characteristic Equation**: p(\lambda) = 0. The solutions are the **Eigenvalues** (characteristic roots).\n* **Eigenvector (X$)**: A non-zero vector that satisfies$A X = \lambda X$, where$\lambda is the associated eigenvalue.\n\n# Questions & Discussion\n\n* **Row Operations and Solutions**: \n    * **Question**: Does multiplying a row by a constant or interchanging rows change the solution?\n    * **Response**: No, the solutions for x$and$y remain the same after these operations.\n    * **Question**: Does adding a constant to every element of a row maintain the solution?\n    * **Response**: No, adding a scalar (e.g., 3$$) to row elements is an illegal operation and changes the system's solution.