// SolveCholesky finds the matrix m that solves A * m = b where A = L * L^T or
// A = U^T * U, and U or L are represented by t, placing the result in the
// receiver.
func (m *Dense) SolveCholesky(t Triangular, b Matrix) {
_, n := t.Dims()
bm, bn := b.Dims()
if n != bm {
panic(ErrShape)
}
m.reuseAs(bm, bn)
if b != m {
m.Copy(b)
}
// TODO(btracey): Implement an algorithm that doesn't require a copy into
// a blas64.Triangular.
ta := getBlasTriangular(t)
switch ta.Uplo {
case blas.Upper:
blas64.Trsm(blas.Left, blas.Trans, 1, ta, m.mat)
blas64.Trsm(blas.Left, blas.NoTrans, 1, ta, m.mat)
case blas.Lower:
blas64.Trsm(blas.Left, blas.NoTrans, 1, ta, m.mat)
blas64.Trsm(blas.Left, blas.Trans, 1, ta, m.mat)
default:
panic(badTriangle)
}
}
// SolveTri finds the matrix x that solves op(A) * X = B where A is a triangular
// matrix and op is specified by trans.
func (m *Dense) SolveTri(a Triangular, trans bool, b Matrix) {
n, _ := a.Triangle()
bm, bn := b.Dims()
if n != bm {
panic(ErrShape)
}
m.reuseAs(bm, bn)
if b != m {
m.Copy(b)
}
// TODO(btracey): Implement an algorithm that doesn't require a copy into
// a blas64.Triangular.
ta := getBlasTriangular(a)
t := blas.NoTrans
if trans {
t = blas.Trans
}
switch ta.Uplo {
case blas.Upper, blas.Lower:
blas64.Trsm(blas.Left, t, 1, ta, m.mat)
default:
panic(badTriangle)
}
}
// SolveCholesky finds the matrix m that solves A * m = b where A is represented
// by the cholesky decomposition, placing the result in the receiver.
func (m *Dense) SolveCholesky(chol *Cholesky, b Matrix) error {
n := chol.chol.mat.N
bm, bn := b.Dims()
if n != bm {
panic(matrix.ErrShape)
}
m.reuseAs(bm, bn)
if b != m {
m.Copy(b)
}
blas64.Trsm(blas.Left, blas.Trans, 1, chol.chol.mat, m.mat)
blas64.Trsm(blas.Left, blas.NoTrans, 1, chol.chol.mat, m.mat)
if chol.cond > matrix.ConditionTolerance {
return matrix.Condition(chol.cond)
}
return nil
}
// SolveCholesky finds the matrix x that solves A * X = B where A = L * L^T or
// A = U^T * U, and U or L are represented by t. The matrix A must be symmetric
// and positive definite.
func (m *Dense) SolveCholesky(t *Triangular, b Matrix) {
_, n := t.Dims()
bm, bn := b.Dims()
if n != bm {
panic(ErrShape)
}
m.reuseAs(bm, bn)
if b != m {
m.Copy(b)
}
switch t.mat.Uplo {
case blas.Upper:
blas64.Trsm(blas.Left, blas.Trans, 1, t.mat, m.mat)
blas64.Trsm(blas.Left, blas.NoTrans, 1, t.mat, m.mat)
case blas.Lower:
blas64.Trsm(blas.Left, blas.NoTrans, 1, t.mat, m.mat)
blas64.Trsm(blas.Left, blas.Trans, 1, t.mat, m.mat)
default:
panic(badTriangle)
}
}
// Solve computes minimum norm least squares solution of a.x = b where b has as many rows as a.
// A matrix x is returned that minimizes the two norm of Q*R*X-B. Solve will panic
// if a is not full rank.
func (f LQFactor) Solve(b *Dense) (x *Dense) {
lq := f.LQ
lDiag := f.lDiag
m, n := lq.Dims()
bm, bn := b.Dims()
if bm != m {
panic(ErrShape)
}
if !f.IsFullRank() {
panic(ErrSingular)
}
x = NewDense(n, bn, nil)
x.Copy(b)
tau := make([]float64, m)
for i := range tau {
tau[i] = lq.at(i, i)
lq.set(i, i, lDiag[i])
}
lqT := blas64.Triangular{
// N omitted since it is not used by Trsm.
Stride: lq.Mat.Stride,
Data: lq.Mat.Data,
Uplo: blas.Lower,
Diag: blas.NonUnit,
}
x.Mat.Rows = bm
blas64.Trsm(blas.Left, blas.NoTrans, 1, lqT, x.Mat)
x.Mat.Rows = n
for i := range tau {
lq.set(i, i, tau[i])
}
f.applyQTo(x, true)
return x
}
// SolveTri finds the matrix x that solves op(A) * X = B where A is a triangular
// matrix and op is specified by trans.
func (m *Dense) SolveTri(a *Triangular, trans bool, b Matrix) {
_, n := a.Dims()
bm, bn := b.Dims()
if n != bm {
panic(ErrShape)
}
m.reuseAs(bm, bn)
if b != m {
m.Copy(b)
}
t := blas.NoTrans
if trans {
t = blas.Trans
}
switch a.mat.Uplo {
case blas.Upper, blas.Lower:
blas64.Trsm(blas.Left, t, 1, a.mat, m.mat)
default:
panic(badTriangle)
}
}
// Solve solves a minimum-norm solution to a system of linear equations defined
// by the matrices a and b. 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.
// 1. If m >= n and trans == false, find X such that ||a*X - b||_2 is minimized.
// 2. If m < n and trans == false, find the minimum norm solution of a * X = b.
// 3. If m >= n and trans == true, find the minimum norm solution of a^T * X = b.
// 4. If m < n and 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) Solve(a, b Matrix) error {
ar, ac := a.Dims()
br, bc := b.Dims()
if ar != br {
panic(ErrShape)
}
m.reuseAs(ac, bc)
// TODO(btracey): Add special cases for SymDense, etc.
aMat, aTrans := untranspose(a)
bMat, bTrans := untranspose(b)
switch rma := aMat.(type) {
case RawTriangular:
side := blas.Left
tA := blas.NoTrans
if aTrans {
tA = blas.Trans
}
if m != bMat {
m.Copy(b)
} else if bTrans {
// m and b share data so Copy cannot be used directly.
tmp := getWorkspace(br, bc, false)
tmp.Copy(b)
m.Copy(tmp)
putWorkspace(tmp)
}
rm := rma.RawTriangular()
blas64.Trsm(side, tA, 1, rm, m.mat)
work := make([]float64, 3*rm.N)
iwork := make([]int, rm.N)
cond := lapack64.Trcon(condNorm, rm, work, iwork)
if cond > condTol {
return Condition(cond)
}
return nil
}
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 m.SolveLU(&lu, false, b)
case ar > ac:
var qr QR
qr.Factorize(a)
return m.SolveQR(&qr, false, b)
default:
var lq LQ
lq.Factorize(a)
return m.SolveLQ(&lq, false, b)
}
}