---
title: Krylov Subspace Methods
description: Krylov subspace methods are iterative algorithms used to find eigenvalues or solve large, sparse systems of linear equations. Learners will understand methods like GMRES and Conjugate Gradient to solve high-dimensional computational problems.
category: mathematics
subcategory: numerical-linear-algebra
difficulty: beginner, intermediate, advanced
url: /subject/krylov-subspace-methods
---

# Krylov Subspace Methods

Krylov subspace methods are iterative algorithms used to find eigenvalues or solve large, sparse systems of linear equations. Learners will understand methods like GMRES and Conjugate Gradient to solve high-dimensional computational problems.

## Available Resources

1 Books • 3 Websites • 1 Papers

## Websites

### 1. 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.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://docs.scipy.org/doc/scipy/reference/sparse.linalg.html

**Tags:** scipy, sparse-linear-algebra, iterative-solvers, krylov-methods, python

### 2. 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.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www-users.cse.umn.edu/~saad/software/

**Tags:** krylov-methods, sparse-matrices, preconditioning, iterative-solvers, numerical-software

### 3. Wolfram MathWorld

**Author:** Eric W. Weisstein

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.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Books

### 1. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods

**Author:** Richard Barrett, Michael Berry, Tony F. Chan, James Demmel, Jack Dongarra, Henk van der Vorst, et al.

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.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.netlib.org/templates/templates.pdf

**Tags:** krylov-methods, iterative-solvers, conjugate-gradient, gmres, preconditioning

## Papers

### 1. An Introduction to the Conjugate Gradient Method Without the Agonizing Pain

**Author:** Jonathan Richard Shewchuk

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.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf

**Tags:** conjugate-gradient, steepest-descent, krylov-methods, preconditioning, numerical-optimization

---

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