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# SparseSolve(_:_:)

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

```
func SparseSolve(_ Factored: SparseOpaqueFactorization_Float, _ XB: DenseMatrix_Float)
```

## 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-3703889@2x.png)

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

```objc
// Create the coefficient matrix `A` from coordinate (COO) arrays.
int rowIndices[]    = {0, 1, 1, 2};
int columnIndices[] = {2, 0, 2, 1};
float aValues[]     = {10, 20, 5, 50};

SparseMatrix_Float A = SparseConvertFromCoordinate(3, 3,
                                                   4, 1,
                                                   (SparseAttributes_t){0},
                                                   rowIndices, columnIndices,
                                                   aValues);

// Compute the factors of `A` using a QR decomposition.
SparseOpaqueFactorization_Float factorization = SparseFactor(SparseFactorizationQR, A);

// Create the right-hand-side matrix `B` (also holds the solution `X` on return).
float bValues[] = {30, 35, 100, 300, 350, 1000};

DenseMatrix_Float XB = {
    .rowCount     = 3,
    .columnCount  = 2,
    .columnStride = 3,
    .attributes   = (SparseAttributes_t){0},
    .data         = bValues
};

// Solve `AX = B` in place.
SparseSolve(factorization, XB);

// Prints "[1, 2, 3, 10, 20, 30]".
printf("[%g, %g, %g, %g, %g, %g]\n",
      bValues[0], bValues[1], bValues[2],
      bValues[3], bValues[4], bValues[5]);

SparseCleanup(A);
SparseCleanup(factorization);
```

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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