DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-30

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Forward and backward substitution for triangular linear systems

Definition

Let $F$ be a field, let $n \geq 1$ , and let $b \in F^{n}$ .

If $L = (ℓ_{i j}) \in M_{n} (F)$ is lower triangular with every diagonal entry $ℓ_{i i} \neq 0$ , the forward substitution solve of $L x = b$ is the recursive construction

$x_{i} : = \frac{1}{ℓ_{i i}} (b_{i} - \sum_{j < i} ℓ_{i j} x_{j}) (0 \leq i < n) .$ If $U = (u_{i j}) \in M_{n} (F)$ is upper triangular with every diagonal entry $u_{i i} \neq 0$ , the backward substitution solve of $U x = b$ is the recursive construction $x_{i} : = \frac{1}{u_{i i}} (b_{i} - \sum_{j > i} u_{i j} x_{j}) (n - 1 \geq i \geq 0) .$

The triangular vocabulary is from Upper triangular, lower triangular and diagonal square matrices over a commutative ring. The matrix product convention from Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes is the one used when later items rewrite these recursions as matrix equations.

Depends on

Used by

Forward and backward substitution are correct, unique, and quadratic in scalar operations Theorem

Dependency tree · two levels

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Sources

Tobin A. Driscoll and Richard J. Braun, Fundamentals of Numerical Computation, Section 2.5.3 (standard reference, not scraped)
David Bindel, CS 4220: Numerical Analysis, Blocked LU and Cholesky (standard reference, not scraped)