// Solve returns a matrix x that satisfies ax = b.
// It returns a nil matrix and ErrSingular if a is singular.
func Solve(a, b Matrix) (x *Dense, err error) {
m, n := a.Dims()
br, _ := b.Dims()
if m != br {
panic("rowMismatch")
}
switch {
case m == n:
var lu LU
lu.Factorize(a)
if lu.Det() == 0 {
return nil, ErrSingular
}
x := DenseCopyOf(b)
lapack64.Getrs(blas.NoTrans, lu.lu.mat, x.mat, lu.pivot)
return x, nil
default:
_, bc := b.Dims()
mn := max(m, n)
// TODO(btracey): Employ special cases to avoid the copy where possible.
aCopy := DenseCopyOf(a)
x := NewDense(mn, bc, nil)
x.Copy(b)
work := make([]float64, 1)
lapack64.Gels(blas.NoTrans, aCopy.mat, x.mat, work, -1)
work = make([]float64, int(work[0]))
ok := lapack64.Gels(blas.NoTrans, aCopy.mat, x.mat, work, len(work))
if !ok {
return nil, ErrSingular
}
return x.View(0, 0, n, bc).(*Dense), nil
}
}
// 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
}
// 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 a test for the condition number of A.
// TODO(btracey): Add special cases for TriDense, SymDense, etc.
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
}
// Solve using an LU decomposition.
var lu LU
lu.Factorize(a)
if lu.Det() == 0 {
return Condition(math.Inf(1))
}
bMat, bTrans := untranspose(b)
if m == bMat && bTrans {
var restore func()
m, restore = m.isolatedWorkspace(bMat)
defer restore()
}
if m != bMat {
m.Copy(b)
}
lapack64.Getrs(blas.NoTrans, lu.lu.mat, m.mat, lu.pivot)
return nil
default:
// Solve using QR/LQ.
// Copy a since the corresponding LAPACK argument is modified during
// the call.
var aCopy Dense
aCopy.Clone(a)
x := getWorkspace(max(ar, ac), bc, false)
defer putWorkspace(x)
x.Copy(b)
work := make([]float64, 1)
lapack64.Gels(blas.NoTrans, aCopy.mat, x.mat, work, -1)
work = make([]float64, int(work[0]))
ok := lapack64.Gels(blas.NoTrans, aCopy.mat, x.mat, work, len(work))
if !ok {
return Condition(math.Inf(1))
}
m.Copy(x)
return nil
}
}