Numerical stability, floating point arithmetic, and iterative solvers for reliable computation.
6 concepts
Floating-point numbers approximate real numbers using a fixed number of bits following the IEEE 754 standard.
Numerical stability measures how much rounding and tiny input changes can distort an algorithm’s output on real computers using floating-point arithmetic.
The Conjugate Gradient (CG) method solves large, sparse, symmetric positive definite (SPD) linear systems Ax = b using only matrix–vector products and dot products.
Matrix factorizations rewrite a matrix into simpler building blocks (triangular or orthogonal) that make solving and analyzing linear systems much easier.
Numerical integration approximates the area under a curve when an exact antiderivative is unknown, using deterministic quadrature rules or random sampling (Monte Carlo).
Numerical differentiation uses finite differences to estimate derivatives when an analytical derivative is hard or impossible to obtain.