// SolveLUVec solves a system of linear equations using the LU decomposition of a matrix.
// It computes
// A * x = b if trans == false
// A^T * x = b if trans == true
// In both cases, A is represeneted in LU factorized form, and the matrix x is
// stored into the receiver.
//
// If A is singular or near-singular a Condition error is returned. Please see
// the documentation for Condition for more information.
func (v *Vector) SolveLUVec(lu *LU, trans bool, b *Vector) error {
_, n := lu.lu.Dims()
bn := b.Len()
if bn != n {
panic(matrix.ErrShape)
}
// TODO(btracey): Should test the condition number instead of testing that
// the determinant is exactly zero.
if lu.Det() == 0 {
return matrix.Condition(math.Inf(1))
}
v.reuseAs(n)
var restore func()
if v == b {
v, restore = v.isolatedWorkspace(b)
defer restore()
}
v.CopyVec(b)
vMat := blas64.General{
Rows: n,
Cols: 1,
Stride: v.mat.Inc,
Data: v.mat.Data,
}
t := blas.NoTrans
if trans {
t = blas.Trans
}
lapack64.Getrs(t, lu.lu.mat, vMat, lu.pivot)
if lu.cond > matrix.ConditionTolerance {
return matrix.Condition(lu.cond)
}
return nil
}
// SolveLU solves a system of linear equations using the LU decomposition of a matrix.
// It computes
// A * x = b if trans == false
// A^T * x = b if trans == true
// In both cases, A is represented in LU factorized form, and the matrix x is
// stored into the receiver.
//
// If A is singular or near-singular a Condition error is returned. Please see
// the documentation for Condition for more information.
func (m *Dense) SolveLU(lu *LU, trans bool, b Matrix) error {
_, n := lu.lu.Dims()
br, bc := b.Dims()
if br != n {
panic(matrix.ErrShape)
}
// TODO(btracey): Should test the condition number instead of testing that
// the determinant is exactly zero.
if lu.Det() == 0 {
return matrix.Condition(math.Inf(1))
}
m.reuseAs(n, bc)
bU, _ := untranspose(b)
var restore func()
if m == bU {
m, restore = m.isolatedWorkspace(bU)
defer restore()
} else if rm, ok := bU.(RawMatrixer); ok {
m.checkOverlap(rm.RawMatrix())
}
m.Copy(b)
t := blas.NoTrans
if trans {
t = blas.Trans
}
lapack64.Getrs(t, lu.lu.mat, m.mat, lu.pivot)
if lu.cond > matrix.ConditionTolerance {
return matrix.Condition(lu.cond)
}
return nil
}
// 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
}
// InverseTri computes the inverse of the triangular matrix a, storing the result
// into the receiver. If a is ill-conditioned, a Condition error will be returned.
// Note that matrix inversion is numerically unstable, and should generally be
// avoided where possible, for example by using the Solve routines.
func (t *TriDense) InverseTri(a Triangular) error {
if rt, ok := a.(RawTriangular); ok {
t.checkOverlap(rt.RawTriangular())
}
n, _ := a.Triangle()
t.reuseAs(a.Triangle())
t.Copy(a)
work := make([]float64, 3*n)
iwork := make([]int, n)
cond := lapack64.Trcon(matrix.CondNorm, t.mat, work, iwork)
if math.IsInf(cond, 1) {
return matrix.Condition(cond)
}
ok := lapack64.Trtri(t.mat)
if !ok {
return matrix.Condition(math.Inf(1))
}
if cond > matrix.ConditionTolerance {
return matrix.Condition(cond)
}
return nil
}
// Inverse computes the inverse of the matrix a, storing the result into the
// receiver. If a is ill-conditioned, a Condition error will be returned.
// Note that matrix inversion is numerically unstable, and should generally
// be avoided where possible, for example by using the Solve routines.
func (m *Dense) Inverse(a Matrix) error {
// TODO(btracey): Special case for RawTriangular, etc.
r, c := a.Dims()
if r != c {
panic(matrix.ErrSquare)
}
m.reuseAs(a.Dims())
aU, aTrans := untranspose(a)
switch rm := aU.(type) {
case RawMatrixer:
if m != aU || aTrans {
if m == aU || m.checkOverlap(rm.RawMatrix()) {
tmp := getWorkspace(r, c, false)
tmp.Copy(a)
m.Copy(tmp)
putWorkspace(tmp)
break
}
m.Copy(a)
}
default:
if m != aU {
m.Copy(a)
} else if aTrans {
// m and a share data so Copy cannot be used directly.
tmp := getWorkspace(r, c, false)
tmp.Copy(a)
m.Copy(tmp)
putWorkspace(tmp)
}
}
ipiv := make([]int, r)
lapack64.Getrf(m.mat, ipiv)
work := make([]float64, 1, 4*r) // must be at least 4*r for cond.
lapack64.Getri(m.mat, ipiv, work, -1)
if int(work[0]) > 4*r {
work = make([]float64, int(work[0]))
} else {
work = work[:4*r]
}
lapack64.Getri(m.mat, ipiv, work, len(work))
norm := lapack64.Lange(matrix.CondNorm, m.mat, work)
cond := lapack64.Gecon(matrix.CondNorm, m.mat, norm, work, ipiv) // reuse ipiv
if cond > matrix.ConditionTolerance {
return matrix.Condition(cond)
}
return nil
}
// 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
}
// SolveCholeskyVec finds the vector v that solves A * v = b where A is represented
// by the Cholesky decomposition, placing the result in the receiver.
func (v *Vector) SolveCholeskyVec(chol *Cholesky, b *Vector) error {
n := chol.chol.mat.N
vn := b.Len()
if vn != n {
panic(matrix.ErrShape)
}
v.reuseAs(n)
if v != b {
v.CopyVec(b)
}
blas64.Trsv(blas.Trans, chol.chol.mat, v.mat)
blas64.Trsv(blas.NoTrans, chol.chol.mat, v.mat)
if chol.cond > matrix.ConditionTolerance {
return matrix.Condition(chol.cond)
}
return nil
}
// Solve finds 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:
// - if m >= n, find X such that ||A*X - B||_2 is minimized,
// - if m < n, find the minimum norm solution of A * X = B.
// 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(matrix.ErrShape)
}
m.reuseAs(ac, bc)
// TODO(btracey): Add special cases for SymDense, etc.
aU, aTrans := untranspose(a)
bU, bTrans := untranspose(b)
switch rma := aU.(type) {
case RawTriangular:
side := blas.Left
tA := blas.NoTrans
if aTrans {
tA = blas.Trans
}
switch rm := bU.(type) {
case RawMatrixer:
if m != bU || bTrans {
if m == bU || m.checkOverlap(rm.RawMatrix()) {
tmp := getWorkspace(br, bc, false)
tmp.Copy(b)
m.Copy(tmp)
putWorkspace(tmp)
break
}
m.Copy(b)
}
default:
if m != bU {
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(matrix.CondNorm, rm, work, iwork)
if cond > matrix.ConditionTolerance {
return matrix.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)
}
}
