Function which returns the least-squares solution to a linear matrix equation

Question

I have been trying to rewrite the code from Python to Swift but I'm stuck on the function which should return the least-squares solution to a linear matrix equation. Does anyone know a library written in Swift which has an equivalent method to the numpy.linalg.lstsq? I'd be grateful for your help.

Python code:

a = numpy.array([[p2.x-p1.x,p2.y-p1.y],[p4.x-p3.x,p4.y-p3.y],[p4.x-p2.x,p4.y-p2.y],[p3.x-p1.x,p3.y-p1.y]])
b = numpy.array([number1,number2,number3,number4])
res = numpy.linalg.lstsq(a,b) 
result = [float(res[0][0]),float(res[0][1])]
return result

Swift code so far:

var matrix1 = [[p2.x-p1.x, p2.y-p1.y],[p4.x-p3.x, p4.y-p3.y], [p4.x-p2.x, p4.y-p2.y], [p3.x-p1.x, p3.y-p1.y]]
var matrix2 = [number1, number2, number3, number4]

Answer 1

The Accelerate framework included the LAPACK linear algebra package, which has a DGELS function to solve under- or overdetermined linear systems. From the documentation:

DGELS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank.

Here is an example how that function can be used from Swift. It is essentially a translation of this C sample code.

func solveLeastSquare(A A: [[Double]], B: [Double]) -> [Double]? {
    precondition(A.count == B.count, "Non-matching dimensions")
    
    var mode = Int8(bitPattern: UInt8(ascii: "N")) // "Normal" mode
    var nrows = CInt(A.count)
    var ncols = CInt(A[0].count)
    var nrhs = CInt(1)
    var ldb = max(nrows, ncols)
    
    // Flattened columns of matrix A
    var localA = (0 ..< nrows * ncols).map {
        A[Int($0 % nrows)][Int($0 / nrows)]
    }

    // Vector B, expanded by zeros if ncols > nrows
    var localB = B
    if ldb > nrows {
        localB.appendContentsOf([Double](count: ldb - nrows, repeatedValue: 0.0))
    }
    
    var wkopt = 0.0
    var lwork: CInt = -1
    var info: CInt = 0
    
    // First call to determine optimal workspace size
    dgels_(&mode, &nrows, &ncols, &nrhs, &localA, &nrows, &localB, &ldb, &wkopt, &lwork, &info)
    lwork = Int32(wkopt)

    // Allocate workspace and do actual calculation
    var work = [Double](count: Int(lwork), repeatedValue: 0.0)
    dgels_(&mode, &nrows, &ncols, &nrhs, &localA, &nrows, &localB, &ldb, &work, &lwork, &info)

    if info != 0 {
        print("A does not have full rank; the least squares solution could not be computed.")
        return nil
    }
    return Array(localB.prefix(Int(ncols)))
}

Some notes:

• dgels_() modifies the passed matrix and vector data, and expects the matrix as "flat" array containing the columns of A. Also the right-hand side is expected as an array with length max(M, N). For this reason, the input data is copied to local variables first.
• All arguments must be passed by reference to dgels_(), that's why they are all stored in vars.
• A C integer is a 32-bit integer, which makes some conversions between Int and CInt necessary.

Example 1: Overdetermined system, from http://www.seas.ucla.edu/~vandenbe/103/lectures/ls.pdf.

let A = [[ 2.0, 0.0 ],
         [ -1.0, 1.0 ],
         [ 0.0, 2.0 ]]
let B = [ 1.0, 0.0, -1.0 ]
if let x = solveLeastSquare(A: A, B: B) {
    print(x) // [0.33333333333333326, -0.33333333333333343]
}

Example 2: Underdetermined system, minimum norm solution to x_1 + x_2 + x_3 = 1.0.

let A = [[ 1.0, 1.0, 1.0 ]]
let B = [ 1.0 ]
if let x = solveLeastSquare(A: A, B: B) {
    print(x) // [0.33333333333333337, 0.33333333333333337, 0.33333333333333337]
}

---

Update for Swift 3 and Swift 4:

func solveLeastSquare(A: [[Double]], B: [Double]) -> [Double]? {
    precondition(A.count == B.count, "Non-matching dimensions")
    
    var mode = Int8(bitPattern: UInt8(ascii: "N")) // "Normal" mode
    var nrows = CInt(A.count)
    var ncols = CInt(A[0].count)
    var nrhs = CInt(1)
    var ldb = max(nrows, ncols)
    
    // Flattened columns of matrix A
    var localA = (0 ..< nrows * ncols).map { (i) -> Double in
        A[Int(i % nrows)][Int(i / nrows)]
    }
    
    // Vector B, expanded by zeros if ncols > nrows
    var localB = B
    if ldb > nrows {
        localB.append(contentsOf: [Double](repeating: 0.0, count: Int(ldb - nrows)))
    }
    
    var wkopt = 0.0
    var lwork: CInt = -1
    var info: CInt = 0
    
    // First call to determine optimal workspace size
    var nrows_copy = nrows // Workaround for SE-0176
    dgels_(&mode, &nrows, &ncols, &nrhs, &localA, &nrows_copy, &localB, &ldb, &wkopt, &lwork, &info)
    lwork = Int32(wkopt)
    
    // Allocate workspace and do actual calculation
    var work = [Double](repeating: 0.0, count: Int(lwork))
    dgels_(&mode, &nrows, &ncols, &nrhs, &localA, &nrows_copy, &localB, &ldb, &work, &lwork, &info)
    
    if info != 0 {
        print("A does not have full rank; the least squares solution could not be computed.")
        return nil
    }
    return Array(localB.prefix(Int(ncols)))
}