##  [Bandwidth](/bandwidth-1) 

 Definition

A measure of how far from the main diagonal the significant nonzero entries of a discretized linear operator (matrix) are concentrated, commonly quantified as the maximum distance |i-j| for which the matrix A(i,j) is nonzero (or the half-bandwidth in symmetric conventions).

 

 

 

 

 

 





## Principle

Principle

The graph and ordering of degrees of freedom determine bandwidth: local stencils or small-element connectivity produce small bandwidth, and bandwidth controls storage layout, fill-in behaviour during factorization, and the complexity of certain direct solvers.

 

 

 

 

 





## Demonstration

Demonstration

A one-dimensional second-order finite-difference discretization with nearest-neighbor coupling yields a tridiagonal matrix with bandwidth 1 (half-bandwidth of 1), while a naive global ordering of a multidimensional mesh can increase bandwidth and thus the cost of LU factorization.

 

 

 

 

## Misapplication

Misapplication

Using bandwidth as the sole predictor of solver cost without accounting for sparsity pattern irregularity, fill-in from factorization, or the effect of reordering and block structure; or optimizing for bandwidth at the expense of parallel communication patterns.

 

 

 

 

 





## Consequence

Consequence

Low bandwidth enables compact banded storage, efficient banded direct solvers, and reduced fill-in; recognizing and reducing bandwidth through reordering (e.g., reverse Cuthill–McKee) can dramatically lower factorization cost for certain problems.

 

 

 

 

## Reversal

Reversal

Full width or global coupling where significant entries exist far from the diagonal, yielding large bandwidth and necessitating dense-storage techniques or factorization with heavy fill-in.

 

 

 

 

 





## Boundary

Boundary

Refers specifically to geometric distance from diagonal in matrix index space and excludes other notions called 'bandwidth' (signal bandwidth, communication bandwidth) unless the context is linear algebraic discretizations.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between bandwidth and more nuanced sparsity metrics (profile, envelope, and graph separator sizes): bandwidth is a simple scalar summary but may hide critical pattern features that determine algorithmic cost.

 

 

 

 

 





## Synthesis

Synthesis

Bandwidth is the diagonal-distance measure of nonzero concentration in a discrete linear operator; it guides storage and solver choices and is reducible by reordering, but must be considered alongside full sparsity structure for accurate performance prediction.