Gauss–Seidel Method

A stationary linear-system iteration that sweeps coordinates in order and immediately reuses each newly computed component within the same sweep.

Version

v3 · 2026-09-19 · History

Domain-specific #

1915

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Numerical Linear Algebra, Stationary Iterative Methods → Mathematics

Aliases

Liebmann Method, Method of Successive Displacement

Core Idea

The Gauss–Seidel method is a stationary iterative solver for a square linear system \(Ax=b\). It visits the component equations in a declared order, solves each equation for its diagonal variable, and immediately uses every newly computed component in the remaining equations of that same sweep.

Write

\[ A=D+L+U, \]

where \(D\) is diagonal and \(L,U\) are the strict lower and upper parts of \(A\). Assuming the diagonal entries are nonzero, the natural-order iteration is

\[ (D+L)x^{(k+1)}=b-Ux^{(k)}, \]

How would you explain it like I'm…

Use the Newest Guess

Imagine a row of friends each guessing a secret number, where each friend's number depends on the others. Going down the line, each friend fixes their guess using the newest guesses from friends who already went. You walk the line again and again until the guesses stop changing much.

Fresh-Answer Equation Sweeps

The Gauss–Seidel Method solves a big set of equations that share unknowns by improving guesses over and over. You go through the equations in a fixed order, and each one is solved for its own unknown. The trick is that as soon as you get a new value, you use it right away in the next equations of the same pass. Older values are used only for unknowns you haven't reached yet in this pass. Repeating these passes moves the guesses toward the answer.

Successive-Update Linear Solver

The Gauss–Seidel Method is an iterative way to solve a square linear system Ax = b. Each sweep visits the equations in a set order and solves equation i for the variable x_i, which sits on the diagonal. When computing x_i, it uses the brand-new values of the variables before it (already updated this sweep) and the previous sweep's values for the variables after it. This differs from the Jacobi method, which updates every variable using only old values, as if all were computed at the same instant. You need the diagonal entries to be nonzero so you can divide by them.

 

Gauss–Seidel is a stationary iterative method for a square linear system Ax = b. Split A = D + L + U into its diagonal, strict lower, and strict upper parts, and assume the diagonal entries are nonzero. The natural-order iteration is (D + L)x^(k+1) = b − Ux^(k), i.e. x^(k+1) = −(D+L)^(-1)Ux^(k) + (D+L)^(-1)b. In practice you never form the inverse; it just denotes a forward (lower-triangular) substitution. Componentwise, x_i^(k+1) = (b_i − Σ_{j<i} a_ij x_j^(k+1) − Σ_{j>i} a_ij x_j^(k)) / a_ii. The mix of superscripts is the defining feature: below-diagonal terms use current-sweep values and above-diagonal terms use last-sweep values. This 'successive displacement' schedule contrasts with Jacobi's, where every update waits on the complete old iterate.

Scope of Application

Gauss–Seidel appears in dense and sparse linear-system solving, finite-difference and finite-element discretizations, elliptic partial differential equations, network equations, and relaxation schemes. It is often more important as a smoother or preconditioning component inside multigrid and Krylov methods than as a standalone high-accuracy solver.

The method is defined for any ordering with usable diagonal pivots, but its convergence is matrix- and ordering-dependent. In exact finite dimensions, convergence from every starting vector occurs exactly when

Clarity

Splitting conventions vary. Some texts write \(A=D-L-U\) with \(L,U\) defined as negatives of the off-diagonal parts. This dossier uses \(A=D+L+U\), so the iteration matrix contains the explicit minus sign. Formulas from different conventions must not be mixed.

Nonzero diagonal entries make component updates executable but do not prove convergence. A valid implementation can faithfully generate a divergent sequence.

Manages Complexity

A direct factorization can require substantial memory and fill for a large sparse matrix. A Gauss–Seidel sweep uses the original sparse coefficients and one mutable vector, making each update local and inexpensive. The lower-triangular interpretation gives a simple reusable kernel.

The method converts one global solve into repeated local equation solves. Fresh values transmit information through a sweep faster than Jacobi's one-step delay, though this often trades away easy parallelism.

Abstract Reasoning

Let \(x^\ast\) solve \(Ax^\ast=b\), and define \(e^{(k)}=x^{(k)}-x^\ast\). Subtracting the exact split equation from the iteration gives

\[ (D+L)e^{(k+1)}=-Ue^{(k)}, \qquad e^{(k+1)}=B_{\mathrm{GS}}e^{(k)}. \]

Therefore \(e^{(k)}=B_{\mathrm{GS}}^ke^{(0)}\). The matrix powers tend to zero for every initial error exactly when the spectral radius is less than one.

Knowledge Transfer

The exact sweep transfers from small hand computations to sparse PDE matrices, block systems, and distributed smoothers. Data structures differ, but the ordered triangular dependency and newest-value reuse must remain.

For symmetric positive-definite \(A\), the system is the stationary condition for

\[ \phi(x)=\frac12x^{\mathsf T}Ax-b^{\mathsf T}x. \]

A coordinate Gauss–Seidel update minimizes this quadratic in one coordinate while holding the others at their newest available values. This supplies a legitimate bridge to coordinate descent, restricted to the quadratic linear-system setting.

Relationships to Other Abstractions

Current abstraction Gauss–Seidel Method Domain-specific

Parents (1) — outgoing typed relations

• Gauss–Seidel Method is a kind of Iteration Prime

  prime:iteration is the broader abstraction this entry instantiates.

Hierarchy path (1) — routes to 1 parentless root

• Gauss–Seidel Method → is a kind of Iteration

Neighborhood in Abstraction Space

Gauss–Seidel Method sits in a sparse region of the domain-specific corpus (88^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Convex Optimization & Iterative Methods (8 abstractions)

Nearest neighbors

• Jacobi Method — 0.86
• Matrix — 0.80
• Drazin inverse — 0.80
• Matrix exponential — 0.80
• Hadamard matrix — 0.79

Computed from structural-signature embeddings · 2026-10-25