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# SparseSolve(_:_:_:_:_:)

Solves the equation *AX = B* for matrices of single-precision values using the specified iterative method and preconditioner type.

```
func SparseSolve(_ method: SparseIterativeMethod, _ A: SparseMatrix_Float, _ B: DenseMatrix_Float, _ X: DenseMatrix_Float, _ Preconditioner: SparsePreconditioner_t) -> SparseIterativeStatus_t
```

## Parameters

`method`

The iterative method.

`A`

The matrix *A*.

`B`

The matrix *B*.

`X`

The matrix *X*.

`Preconditioner`

The preconditioner to apply.

## Return Value

A [`SparseIterativeStatus_t`](/documentation/Accelerate/SparseIterativeStatus_t) enumeration that represents the status of the iterative solve.

## Discussion

Use this function to solve a system of linear equations using a factored coefficient matrix. Preconditioning the coefficient matrix can reduce the number of iterations the function requires to converge the system.

The following figure shows two systems of equations where the coefficient matrix is sparse:

![A mathematical equation that has two stacked sets of three simultaneous equations on the left. Each equation has three unknowns. The same sets of simultaneous equations appear on the right as a single matrix equation, A x equals B. The single matrix equation consists of a three-by-three matrix multiplied by a three-by-two matrix that equals a three-by-two matrix. ](images/com.apple.accelerate/media-3703920@2x.png)

The following code solves this system by applying a diagonal scaling preconditioner and using the least squares minimum residual method:

```swift
/// Create the coefficient matrix _A_.
let rowIndices: [Int32] =    [ 0,  1, 1,  2]
let columnIndices: [Int32] = [ 2,  0, 2,  1]
let aValues: [Float] =       [10, 20, 5, 50]

let A = SparseConvertFromCoordinate(3, 3,
                                    4, 1,
                                    SparseAttributes_t(),
                                    rowIndices, columnIndices,
                                    aValues)

defer {
    SparseCleanup(A)
}

/// Create the right-hand-side matrix, _B_.
var bValues: [Float] = [30, 35, 100,
                        300, 350, 1000]
let n = bValues.count

let xValues = [Float](unsafeUninitializedCapacity: n) {
    buffer, count in
    bValues.withUnsafeMutableBufferPointer { bPtr in
        let B = DenseMatrix_Float(rowCount: 3,
                                  columnCount: 2,
                                  columnStride: 3,
                                  attributes: SparseAttributes_t(),
                                  data: bPtr.baseAddress!)
        
        let X = DenseMatrix_Float(rowCount: 3,
                                  columnCount: 2,
                                  columnStride: 3,
                                  attributes: SparseAttributes_t(),
                                  data: buffer.baseAddress!)
        
        SparseSolve(SparseLSMR(),
                    A, B, X,
                    SparsePreconditionerDiagScaling)
        count = n
    }
}
```

On return, x`Values` contains the values `[1.0, 2.0, 3.0, 10.0, 20.0, 30.0]`.

---

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