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Linear Problems

LinearProblem is the SciMLBase problem type for linear systems, re-exported by LinearSolve.jl. Its docstring is maintained in SciMLBase and rendered in the SciMLBase problem interface documentation; this page summarizes how LinearSolve.jl uses it.

Mathematical Specification

A LinearProblem defines the system

\[Au = b\]

where A is either a concrete AbstractMatrix (dense, sparse, GPU, ...) or a matrix-free operator from the SciMLOperators interface, and b is the right-hand side. Matrix-free operators are not compatible with every solver; a solver's needs_concrete_A(alg) reports whether it requires a materialized matrix.

Constructors

LinearProblem(A, b, p = NullParameters(); u0 = nothing, kwargs...)
LinearProblem{isinplace}(A, b, p = NullParameters(); u0 = nothing, kwargs...)
LinearProblem(f::AbstractSciMLOperator, b, p = NullParameters(); u0 = nothing, kwargs...)
  • A, b: the operator and right-hand side.
  • p: problem parameters, NullParameters() when absent (currently unused by the solvers).
  • u0: an optional initial guess, used by iterative methods.
  • isinplace: whether solvers may mutate A and b; defaults to true for an AbstractMatrix and follows the operator's own setting for an AbstractSciMLOperator.
  • kwargs: any extra keyword arguments are stored on the problem and passed to solve; see Common Solver Options.

Fields

  • A: the linear operator.
  • b: the right-hand side.
  • p: the parameters.
  • u0: the initial guess, or nothing.
  • symbolic_interface: a SymbolicLinearInterface when the problem was built by a symbolic front end, else nothing.
  • kwargs: keyword arguments forwarded to the solvers.

Example

using LinearSolve

A = rand(4, 4)
b = rand(4)
prob = LinearProblem(A, b)
sol = solve(prob)
sol.u