mat/solve.go

// Copyright ©2015 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package mat

// Solve solves the linear least squares problem
//
//	minimize over x |b - A*x|_2
//
// where A is an m×n matrix, b is a given m element vector and x is n element
// solution vector. Solve assumes that A has full rank, that is
//
//	rank(A) = min(m,n)
//
// If m >= n, Solve finds the unique least squares solution of an overdetermined
// system.
//
// If m < n, there is an infinite number of solutions that satisfy b-A*x=0. In
// this case Solve finds the unique solution of an underdetermined system that
// minimizes |x|_2.
//
// Several right-hand side vectors b and solution vectors x can be handled in a
// single call. Vectors b are stored in the columns of the m×k matrix B. Vectors
// x will be stored in-place into the n×k receiver.
//
// If the underlying matrix of a is a SolveToer, its SolveTo method is used,
// otherwise a Dense copy of a will be used for the solution.
//
// If A does not have full rank, a Condition error is returned. See the
// documentation for Condition for more information.
func (m *Dense) Solve(a, b Matrix) error {
	aU, aTrans := untransposeExtract(a)
	if a, ok := aU.(SolveToer); ok {
		return a.SolveTo(m, aTrans, b)
	}

	ar, ac := a.Dims()
	br, bc := b.Dims()
	if ar != br {
		panic(ErrShape)
	}
	m.reuseAsNonZeroed(ac, bc)

	switch {
	case ar == ac:
		if a == b {
			// x = I.
			if ar == 1 {
				m.mat.Data[0] = 1
				return nil
			}
			for i := 0; i < ar; i++ {
				v := m.mat.Data[i*m.mat.Stride : i*m.mat.Stride+ac]
				zero(v)
				v[i] = 1
			}
			return nil
		}
		var lu LU
		lu.Factorize(a)
		return lu.SolveTo(m, false, b)
	case ar > ac:
		var qr QR
		qr.Factorize(a)
		return qr.SolveTo(m, false, b)
	default:
		var lq LQ
		lq.Factorize(a)
		return lq.SolveTo(m, false, b)
	}
}

// SolveVec solves the linear least squares problem
//
//	minimize over x |b - A*x|_2
//
// where A is an m×n matrix, b is a given m element vector and x is n element
// solution vector. Solve assumes that A has full rank, that is
//
//	rank(A) = min(m,n)
//
// If m >= n, Solve finds the unique least squares solution of an overdetermined
// system.
//
// If m < n, there is an infinite number of solutions that satisfy b-A*x=0. In
// this case Solve finds the unique solution of an underdetermined system that
// minimizes |x|_2.
//
// The solution vector x will be stored in-place into the receiver.
//
// If A does not have full rank, a Condition error is returned. See the
// documentation for Condition for more information.
func (v *VecDense) SolveVec(a Matrix, b Vector) error {
	if _, bc := b.Dims(); bc != 1 {
		panic(ErrShape)
	}
	_, c := a.Dims()

	// The Solve implementation is non-trivial, so rather than duplicate the code,
	// instead recast the VecDenses as Dense and call the matrix code.

	if rv, ok := b.(RawVectorer); ok {
		bmat := rv.RawVector()
		if v != b {
			v.checkOverlap(bmat)
		}
		v.reuseAsNonZeroed(c)
		m := v.asDense()
		// We conditionally create bm as m when b and v are identical
		// to prevent the overlap detection code from identifying m
		// and bm as overlapping but not identical.
		bm := m
		if v != b {
			b := VecDense{mat: bmat}
			bm = b.asDense()
		}
		return m.Solve(a, bm)
	}

	v.reuseAsNonZeroed(c)
	m := v.asDense()
	return m.Solve(a, b)
}