SparseSolve(_:_:_:_:)
Solves the system Ax = b using the supplied double-precision factorization of A, without any internal memory allocations.
Declaration
func SparseSolve(_ Factored: SparseOpaqueFactorization_Double, _ b: DenseVector_Double, _ x: DenseVector_Double, _ workspace: UnsafeMutableRawPointer)Parameters
- Factored:
The factored matrix to solve.
- b:
The vector b.
- x:
The vector x.
- workspace:
The scratch space of size Solveworkspacerequiredstatic
+ nrhs *Solveworkspacerequiredperrhs.
Discussion
Use this function to solve a system of linear equations using a factored coefficient matrix. In cases where your code calls the function frequently, create and manage the workspace that the Sparse Solvers library uses and reuse it across function calls. Reusing a workspace prevents the Sparse Solvers library from allocating the temporary storage with each call.
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: [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 workspace.
let byteCount = factorization.solveWorkspaceRequiredStatic +
factorization.solveWorkspaceRequiredPerRHS
let workspace = UnsafeMutableRawPointer.allocate(
byteCount: byteCount,
alignment: MemoryLayout<Double>.alignment)
defer {
workspace.deallocate()
}
/// Create the right-hand-side vector, _b_.
var bValues: [Double] = [30, 35, 100]
let n = bValues.count
/// Solve the system.
let xValues = [Double](unsafeUninitializedCapacity: n) {
buffer, count in
bValues.withUnsafeMutableBufferPointer { bPtr in
let b = DenseVector_Double(count: 3,
data: bPtr.baseAddress!)
let x = DenseVector_Double(count: 3,
data: buffer.baseAddress!)
SparseSolve(factorization, b, x,
workspace)
count = n
}
}On return, xValues contains the values [1.0, 2.0, 3.0].
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.