// SolveQR finds a minimum-norm solution to a system of linear equations defined
// by the matrices A and b, where A is an m×n matrix represented in its QR factorized
// form. If A is singular or near-singular a Condition error is returned. Please
// see the documentation for Condition for more information.
//
// The minimization problem solved depends on the input parameters.
// If trans == false, find X such that ||A*X - b||_2 is minimized.
// If trans == true, find the minimum norm solution of A^T * X = b.
// The solution matrix, X, is stored in place into the receiver.
func (m *Dense) SolveQR(qr *QR, trans bool, b Matrix) error {
r, c := qr.qr.Dims()
br, bc := b.Dims()
// The QR solve algorithm stores the result in-place into the right hand side.
// The storage for the answer must be large enough to hold both b and x.
// However, this method's receiver must be the size of x. Copy b, and then
// copy the result into m at the end.
if trans {
if c != br {
panic(matrix.ErrShape)
}
m.reuseAs(r, bc)
} else {
if r != br {
panic(matrix.ErrShape)
}
m.reuseAs(c, bc)
}
// Do not need to worry about overlap between m and b because x has its own
// independent storage.
x := getWorkspace(max(r, c), bc, false)
x.Copy(b)
t := qr.qr.asTriDense(qr.qr.mat.Cols, blas.NonUnit, blas.Upper).mat
if trans {
ok := lapack64.Trtrs(blas.Trans, t, x.mat)
if !ok {
return matrix.Condition(math.Inf(1))
}
for i := c; i < r; i++ {
zero(x.mat.Data[i*x.mat.Stride : i*x.mat.Stride+bc])
}
work := make([]float64, 1)
lapack64.Ormqr(blas.Left, blas.NoTrans, qr.qr.mat, qr.tau, x.mat, work, -1)
work = make([]float64, int(work[0]))
lapack64.Ormqr(blas.Left, blas.NoTrans, qr.qr.mat, qr.tau, x.mat, work, len(work))
} else {
work := make([]float64, 1)
lapack64.Ormqr(blas.Left, blas.Trans, qr.qr.mat, qr.tau, x.mat, work, -1)
work = make([]float64, int(work[0]))
lapack64.Ormqr(blas.Left, blas.Trans, qr.qr.mat, qr.tau, x.mat, work, len(work))
ok := lapack64.Trtrs(blas.NoTrans, t, x.mat)
if !ok {
return matrix.Condition(math.Inf(1))
}
}
// M was set above to be the correct size for the result.
m.Copy(x)
putWorkspace(x)
if qr.cond > matrix.ConditionTolerance {
return matrix.Condition(qr.cond)
}
return nil
}
// QFromQR extracts the m×m orthonormal matrix Q from a QR decomposition.
func (m *Dense) QFromQR(qr *QR) {
r, _ := qr.qr.Dims()
m.reuseAsZeroed(r, r)
// Set Q = I.
for i := 0; i < r*r; i += r + 1 {
m.mat.Data[i] = 1
}
// Construct Q from the elementary reflectors.
work := make([]float64, 1)
lapack64.Ormqr(blas.Left, blas.NoTrans, qr.qr.mat, qr.tau, m.mat, work, -1)
work = make([]float64, int(work[0]))
lapack64.Ormqr(blas.Left, blas.NoTrans, qr.qr.mat, qr.tau, m.mat, work, len(work))
}