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# Sparse Iterative Methods

Select a suitable iterative method to solve a system.

## Discussion

Sparse Iterative methods solve *Ax = b* through an iterative process that only requires multiplication by *A* or *A__ᵀ*. However, if *A* is numerically difficult, the iterative process may fail to converge to a solution. Even for problems where the process converges, it may do so slowly. You can fix both of these issues through the application of a problem-specific preconditioner that approximates the inverse of *A*.

## Topics

### Sparse Iterative Methods for Symmetric Positive-Definite Coefficient Matrices

[`SparseConjugateGradient`](/documentation/Accelerate/SparseConjugateGradient())

Returns a conjugate gradient (CG) method.

[`SparseConjugateGradient`](/documentation/Accelerate/SparseConjugateGradient(_:))

Returns a conjugate gradient (CG) method with specified options.

[`SparseCGOptions`](/documentation/Accelerate/SparseCGOptions)

Options for creating a conjugate gradient (CG) method.

### Sparse Iterative Methods for Symmetric Indefinite and Unsymmetric Coefficient Matrices

[`SparseGMRES`](/documentation/Accelerate/SparseGMRES())

Returns a generalized minimal residual (GMRES) method.

[`SparseGMRES`](/documentation/Accelerate/SparseGMRES(_:))

Returns a generalized minimal residual (GMRES) method with specified options.

[`SparseGMRESOptions`](/documentation/Accelerate/SparseGMRESOptions)

Options for creating a generalized minimal residual (GMRES) method.

### Sparse Iterative Methods for Overdetermined and Underdetermined Systems

[`SparseLSMR`](/documentation/Accelerate/SparseLSMR())

Returns a default least squares minimum residual (LSMR) method.

[`SparseLSMR`](/documentation/Accelerate/SparseLSMR(_:))

Returns a least squares minimum residual method with specified options.

[`SparseLSMROptions`](/documentation/Accelerate/SparseLSMROptions)

Options for creating a least squares minimum residual method.

### Iterative Method Structure

[`SparseIterativeMethod`](/documentation/Accelerate/SparseIterativeMethod)

The base type for all iterative methods.


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