An Introduction to the Conjugate Gradient Method Without the Agonizing Pain
by Jonathan Richard Shewchuk · Carnegie Mellon University
A 64-page tutorial deriving Steepest Descent, Conjugate Directions and Conjugate Gradients from quadratic forms, with 66 figures. Uses eigenvector analysis to explain convergence rates, then extends to preconditioning and the nonlinear Conjugate Gradient method.
More resources on Krylov Subspace Methods
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
SciPy Sparse Linear Algebra
Official SciPy reference for the scipy.sparse.linalg module, documenting Krylov solvers such as CG, GMRES, BiCGSTAB, MINRES and LSQR, plus LinearOperator and incomplete-LU preconditioners. Readers can solve large sparse linear systems and eigenproblems in Python.
Yousef Saad's Software Page
Software page of Yousef Saad, author of Iterative Methods for Sparse Linear Systems, listing his research codes such as SPARSKIT, ITSOL and EVSL. Readers can study and run reference implementations of Krylov solvers, preconditioners and sparse eigenvalue methods.
Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods
Algorithm-first reference presenting each iterative method — CG, GMRES, BiCG, QMR, Chebyshev — as compact pseudocode with notes on convergence, stopping criteria and preconditioner choice. Includes a flowchart for selecting a method from matrix properties.