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# SparseSolve(_:_:)

Solves the system *AX = B* using the supplied double-precision factorization of *A*, in place.

```
func SparseSolve(_ Factored: SparseOpaqueFactorization_Double, _ XB: DenseMatrix_Double)
```

## Parameters

`Factored`

The factorization of *A*.

`XB`

On entry, the right-hand-side, *B*. On return, the solution vectors *X*. If *A* has dimension *m x n*, *XB* must have dimension *k x nrhs*, where *k = max(m,n)* and *nrhs* is the number of right-hand-sides to find solutions for.

## Discussion

Use this function to solve a system of linear equations using a factored coefficient matrix. 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-3703886@2x.png)

The following code solves this system with a QR factorization of the coefficient matrix:

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

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

/// Factorize _A_.
let factorization = SparseFactor(SparseFactorizationQR, A)

defer {
    SparseCleanup(A)
    SparseCleanup(factorization)
}

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

/// Solve the system.
bValues.withUnsafeMutableBufferPointer { bPtr in
    let XB = DenseMatrix_Double(rowCount: 3,
                                columnCount: 2,
                                columnStride: 3,
                                attributes: SparseAttributes_t(),
                                data: bPtr.baseAddress!)
    
    SparseSolve(factorization, XB)
}
```

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

If the factorization is *A = QR*, the function returns the solution of minimum norm *‖ x ‖₂* for underdetermined systems.

If the factorization is *A = QR*, the function returns the least squares solution *minₓ ‖ AX - B ‖₂* for overdetermined systems.

If the factorization is [`SparseFactorizationCholeskyAtA`](/documentation/Accelerate/SparseFactorizationCholeskyAtA), the factorization is of *AᵀA*, and the solution that returns is for the system *AᵀAX = B*.

---

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