SparseSolve(_:_:_:)
Solves the system AX = B using the supplied single-precision factorization of A.
Declaration
func SparseSolve(_ Factored: SparseOpaqueFactorization_Float, _ B: DenseMatrix_Float, _ X: DenseMatrix_Float)Parameters
- Factored:
The factorization of A.
- B:
The right-hand-side, B.
- X:
The matrix for returning solutions.
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:
[Image]
The following code solves this system with a QR factorization of the coefficient matrix:
/// 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)
/// Factorize _A_.
let factorization = SparseFactor(SparseFactorizationQR, A)
defer {
SparseCleanup(A)
SparseCleanup(factorization)
}
/// Create the right-hand-side matrix, _b_.
var bValues: [Float] = [30, 35, 100,
300, 350, 1000]
let n = bValues.count
/// Solve the system.
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(factorization, B, X)
count = n
}
}On return, xValues 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, the factorization is of AᵀA, and the solution that returns is for the system AᵀAX = B.