// SolveLQ 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 LQ 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 the minimum norm solution of A * X = b.
// If trans == true, find X such that ||A*X - b||_2 is minimized.
// The solution matrix, X, is stored in place into the receiver.
func (m *Dense) SolveLQ(lq *LQ, trans bool, b Matrix) error {
r, c := lq.lq.Dims()
br, bc := b.Dims()
// The LQ 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 := lq.lq.asTriDense(lq.lq.mat.Rows, blas.NonUnit, blas.Lower).mat
if trans {
work := make([]float64, 1)
lapack64.Ormlq(blas.Left, blas.NoTrans, lq.lq.mat, lq.tau, x.mat, work, -1)
work = make([]float64, int(work[0]))
lapack64.Ormlq(blas.Left, blas.NoTrans, lq.lq.mat, lq.tau, x.mat, work, len(work))
ok := lapack64.Trtrs(blas.Trans, t, x.mat)
if !ok {
return matrix.Condition(math.Inf(1))
}
} else {
ok := lapack64.Trtrs(blas.NoTrans, t, x.mat)
if !ok {
return matrix.Condition(math.Inf(1))
}
for i := r; i < c; i++ {
zero(x.mat.Data[i*x.mat.Stride : i*x.mat.Stride+bc])
}
work := make([]float64, 1)
lapack64.Ormlq(blas.Left, blas.Trans, lq.lq.mat, lq.tau, x.mat, work, -1)
work = make([]float64, int(work[0]))
lapack64.Ormlq(blas.Left, blas.Trans, lq.lq.mat, lq.tau, x.mat, work, len(work))
}
// M was set above to be the correct size for the result.
m.Copy(x)
putWorkspace(x)
if lq.cond > matrix.ConditionTolerance {
return matrix.Condition(lq.cond)
}
return nil
}
// SymRankOne performs a rank-1 update of the original matrix A and refactorizes
// its Cholesky factorization, storing the result into the reciever. That is, if
// in the original Cholesky factorization
// U^T * U = A,
// in the updated factorization
// U'^T * U' = A + alpha * x * x^T = A'.
//
// Note that when alpha is negative, the updating problem may be ill-conditioned
// and the results may be inaccurate, or the updated matrix A' may not be
// positive definite and not have a Cholesky factorization. SymRankOne returns
// whether the updated matrix A' is positive definite.
//
// SymRankOne updates a Cholesky factorization in O(n²) time. The Cholesky
// factorization computation from scratch is O(n³).
func (c *Cholesky) SymRankOne(orig *Cholesky, alpha float64, x *Vector) (ok bool) {
if !orig.valid() {
panic(badCholesky)
}
n := orig.Size()
if x.Len() != n {
panic(matrix.ErrShape)
}
if orig != c {
if c.isZero() {
c.chol = NewTriDense(n, matrix.Upper, nil)
} else if c.chol.mat.N != n {
panic(matrix.ErrShape)
}
c.chol.Copy(orig.chol)
}
if alpha == 0 {
return true
}
// Algorithms for updating and downdating the Cholesky factorization are
// described, for example, in
// - J. J. Dongarra, J. R. Bunch, C. B. Moler, G. W. Stewart: LINPACK
// Users' Guide. SIAM (1979), pages 10.10--10.14
// or
// - P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders: Methods for
// modifying matrix factorizations. Mathematics of Computation 28(126)
// (1974), Method C3 on page 521
//
// The implementation is based on LINPACK code
// http://www.netlib.org/linpack/dchud.f
// http://www.netlib.org/linpack/dchdd.f
// and
// https://icl.cs.utk.edu/lapack-forum/viewtopic.php?f=2&t=2646
//
// According to http://icl.cs.utk.edu/lapack-forum/archives/lapack/msg00301.html
// LINPACK is released under BSD license.
//
// See also:
// - M. A. Saunders: Large-scale Linear Programming Using the Cholesky
// Factorization. Technical Report Stanford University (1972)
// http://i.stanford.edu/pub/cstr/reports/cs/tr/72/252/CS-TR-72-252.pdf
// - Matthias Seeger: Low rank updates for the Cholesky decomposition.
// EPFL Technical Report 161468 (2004)
// http://infoscience.epfl.ch/record/161468
work := make([]float64, n)
blas64.Copy(n, x.RawVector(), blas64.Vector{1, work})
if alpha > 0 {
// Compute rank-1 update.
if alpha != 1 {
blas64.Scal(n, math.Sqrt(alpha), blas64.Vector{1, work})
}
umat := c.chol.mat
stride := umat.Stride
for i := 0; i < n; i++ {
// Compute parameters of the Givens matrix that zeroes
// the i-th element of x.
c, s, r, _ := blas64.Rotg(umat.Data[i*stride+i], work[i])
if r < 0 {
// Multiply by -1 to have positive diagonal
// elemnts.
r *= -1
c *= -1
s *= -1
}
umat.Data[i*stride+i] = r
if i < n-1 {
// Multiply the extended factorization matrix by
// the Givens matrix from the left. Only
// the i-th row and x are modified.
blas64.Rot(n-i-1,
blas64.Vector{1, umat.Data[i*stride+i+1 : i*stride+n]},
blas64.Vector{1, work[i+1 : n]},
c, s)
}
}
c.updateCond(-1)
return true
}
// Compute rank-1 downdate.
alpha = math.Sqrt(-alpha)
if alpha != 1 {
blas64.Scal(n, alpha, blas64.Vector{1, work})
}
// Solve U^T * p = x storing the result into work.
ok = lapack64.Trtrs(blas.Trans, c.chol.RawTriangular(), blas64.General{
Rows: n,
Cols: 1,
Stride: 1,
Data: work,
})
if !ok {
// The original matrix is singular. Should not happen, because
// the factorization is valid.
panic(badCholesky)
}
norm := blas64.Nrm2(n, blas64.Vector{1, work})
if norm >= 1 {
// The updated matrix is not positive definite.
return false
}
norm = math.Sqrt((1 + norm) * (1 - norm))
cos := make([]float64, n)
sin := make([]float64, n)
for i := n - 1; i >= 0; i-- {
// Compute parameters of Givens matrices that zero elements of p
// backwards.
cos[i], sin[i], norm, _ = blas64.Rotg(norm, work[i])
if norm < 0 {
norm *= -1
cos[i] *= -1
sin[i] *= -1
}
}
umat := c.chol.mat
stride := umat.Stride
for i := n - 1; i >= 0; i-- {
// Apply Givens matrices to U.
// TODO(vladimir-ch): Use workspace to avoid modifying the
// receiver in case an invalid factorization is created.
blas64.Rot(n-i, blas64.Vector{1, work[i:n]}, blas64.Vector{1, umat.Data[i*stride+i : i*stride+n]}, cos[i], sin[i])
if umat.Data[i*stride+i] == 0 {
// The matrix is singular (may rarely happen due to
// floating-point effects?).
ok = false
} else if umat.Data[i*stride+i] < 0 {
// Diagonal elements should be positive. If it happens
// that on the i-th row the diagonal is negative,
// multiply U from the left by an identity matrix that
// has -1 on the i-th row.
blas64.Scal(n-i, -1, blas64.Vector{1, umat.Data[i*stride+i : i*stride+n]})
}
}
if ok {
c.updateCond(-1)
} else {
c.Reset()
}
return ok
}
