Resources
- Jacob Bishop’s Tutorials Part 5-7 covers most topics
- Gauss Quadrature Vs Newton Cotes’ Visualization
- Check Calculator function input for Iterative Approximation Methods( Newton-Rhapson, Fixed Point Iteration etc.)
Basic Analysis
- Catastrophic Cancellation
- Significant Digits Vs Decimal points
- Sum for minimal error, rounding/chopping
- Taylor Series’ Approximation and Error
Iterative Methods
- Bisection
- Regula Falsi (Method of Chords)
- Secant
- Swap to whichever is close
- Fixed Point Iteration
- Order of convergences
- Uniqueness
- Error Analysis
- Newton Raphson
- Order of convergences
Matrix Techniques
- Pivoting (Partial Pivoting = Every Step)
- Scaling
- First before Pivoting
- Matrix Norms
- Norm_1(A) = max(col sum)
- Norm_inf(A) = max(row sum)
- Norm_2(A) = Spectral Norm = $\sqrt \lambda$ for eigenvalue
- Norm_F(A) = root of(Sum of all a_ij^2) = Frobenius Norm
- Conditional No. = ||A|| * ||A^-1||
Matrix Iterative Methods
In order to solve, systems of equation we have Gauss-El-M, Gauss-Jacobi and Gauss-Siedel. Jacobi and Siedel is mostly used for sparse arrays where G-El-M is highly ineffecient. Newton-Raphson, Fixed Point are iterative methods also work while finding solution.
- Gaussian Elimination
- G-Jacobi’s
- G-Jacobi’s Matrix Version
- G-Siedel
- G-Siedel Matrix Version
- Fixed Point (Matrix)
- Newton-Raphson (Matrix)
Points to note
- Matrix must be Diagonally Dominant (see thm) for Jacobi/Gauss-Siedel
Interpolation Methods
- Polynomial
- Lagrange
- Error
- Newton Divided Difference
- Newton Forward Difference*
- Newton Backward Difference*
- Hermite
- Oscullatory (generalized Hermite)
- NBD, NFD is only for equally spaced
Table for N.D.D-like Interpolation
- NDD : $frac{f_2 - f_1}{ x_2 - x_1 }$ for each el
- NFD: $f_2 -f_1$ Only
- NBD: Start Reverse and Take Bottom Row; Same table as NFD $f_2 - f_1$;
- HERM: Repeat each entry twice; $frac{f_2 - f_1}{x_2 - x_1} if different or differentiation if same
- OSC: Repeat based on no. of available vals in x, y, y’ ..; $f_2 - f_1$
Approximation for Integration
In order of specific to generalized.
- Newton Cotes’ Formula - Approximate function as polynomial and find area.
- Trapezoid Rule (n = 1 approx:2 data points linear Pn)
- Simpson’s 1/3 Rule (n = 2 approx:3 data points quadratic Pn)
- Simpson’s 3/8 Rule (n = 3 approx:4 data points cubic Pn)
- Generalized Newton Cotes'
- Gaussian Quadrature Method (Method of Undetermined Coeffecients) - Choose points that make the equivalent polygon (may not be contained)
- Gauss-Chebyshev, Gauss-Legendre, Gauss-Hermite with different weight functions
Approximations for Initial Value Problems
Use Taylor Series to derive the equations! Useful predictor methods also,
- Euler’s Method (RK 1st Order)
- Modified Euler Method (RK with 2nd Order AKA Heun’s Method)
- Runge Kutta(RK) Schemes: Move along the slope(which is approximated by the order) from one point to the next point
- Taylor Series Expansion
See Use of Butcher Table also and RK in 2 variable (using 2 variable Taylor Series)!
Multi-Step Method
- Predictor: Adams-Bashforth (Derived from previous m+1 points NBD Polynomial approx)
- Corrector: Adams-Moulton (Derived from previous(and including) m +2 points NBD Polynomial)
Milne’s Predictor Formula (Generalized Adams-Besforth by changing limits of integration)
Approximating System of ODEs
- Runge-Kutta 2 variable method
- Implicit Euler’s
- Milne’s Method
Approximations for Boundary Value Problems
- Finite Difference Method
- Finite Element Methods
- Collocation Method
- Rayleigh Ritz Method
- Galerekin’s Weighting Function Method