// 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(ErrSquare)
}
m.reuseAs(a.Dims())
aMat, aTrans := untranspose(a)
if m != aMat {
m.Copy(a)
} else if aTrans {
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(condNorm, m.mat, work)
cond := lapack64.Gecon(condNorm, m.mat, norm, work, ipiv) // reuse ipiv
if cond > condTol {
return 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
}
// updateCond updates the stored condition number of the matrix. Norm is the
// norm of the original matrix. If norm is negative it will be estimated.
func (lu *LU) updateCond(norm float64) {
n := lu.lu.mat.Cols
work := make([]float64, 4*n)
iwork := make([]int, n)
if norm < 0 {
// This is an approximation. By the definition of a norm, ||AB|| <= ||A|| ||B||.
// The condition number is ||A|| || A^-1||, so this will underestimate
// the condition number somewhat.
// The norm of the original factorized matrix cannot be stored because of
// update possibilities, e.g. RankOne.
u := lu.lu.asTriDense(n, blas.NonUnit, blas.Upper)
l := lu.lu.asTriDense(n, blas.Unit, blas.Lower)
unorm := lapack64.Lantr(matrix.CondNorm, u.mat, work)
lnorm := lapack64.Lantr(matrix.CondNorm, l.mat, work)
norm = unorm * lnorm
}
v := lapack64.Gecon(matrix.CondNorm, lu.lu.mat, norm, work, iwork)
lu.cond = 1 / v
}
// Cond returns the condition number of the given matrix under the given norm.
// The condition number must be based on the 1-norm, 2-norm or ∞-norm.
// Cond will panic with matrix.ErrShape if the matrix has zero size.
//
// BUG(btracey): The computation of the 1-norm and ∞-norm for non-square matrices
// is innacurate, although is typically the right order of magnitude. See
// https://github.com/xianyi/OpenBLAS/issues/636. While the value returned will
// change with the resolution of this bug, the result from Cond will match the
// condition number used internally.
func Cond(a Matrix, norm float64) float64 {
m, n := a.Dims()
if m == 0 || n == 0 {
panic(matrix.ErrShape)
}
var lnorm lapack.MatrixNorm
switch norm {
default:
panic("mat64: bad norm value")
case 1:
lnorm = lapack.MaxColumnSum
case 2:
var svd SVD
ok := svd.Factorize(a, matrix.SVDNone)
if !ok {
return math.Inf(1)
}
return svd.Cond()
case math.Inf(1):
lnorm = lapack.MaxRowSum
}
if m == n {
// Use the LU decomposition to compute the condition number.
tmp := getWorkspace(m, n, false)
tmp.Copy(a)
work := make([]float64, 4*n)
aNorm := lapack64.Lange(lnorm, tmp.mat, work)
pivot := make([]int, m)
lapack64.Getrf(tmp.mat, pivot)
iwork := make([]int, n)
v := lapack64.Gecon(lnorm, tmp.mat, aNorm, work, iwork)
putWorkspace(tmp)
return 1 / v
}
if m > n {
// Use the QR factorization to compute the condition number.
tmp := getWorkspace(m, n, false)
tmp.Copy(a)
work := make([]float64, 3*n)
tau := make([]float64, min(m, n))
lapack64.Geqrf(tmp.mat, tau, work, -1)
if int(work[0]) > len(work) {
work = make([]float64, int(work[0]))
}
lapack64.Geqrf(tmp.mat, tau, work, len(work))
iwork := make([]int, n)
r := tmp.asTriDense(n, blas.NonUnit, blas.Upper)
v := lapack64.Trcon(lnorm, r.mat, work, iwork)
putWorkspace(tmp)
return 1 / v
}
// Use the LQ factorization to compute the condition number.
tmp := getWorkspace(m, n, false)
tmp.Copy(a)
work := make([]float64, 3*m)
tau := make([]float64, min(m, n))
lapack64.Gelqf(tmp.mat, tau, work, -1)
if int(work[0]) > len(work) {
work = make([]float64, int(work[0]))
}
lapack64.Gelqf(tmp.mat, tau, work, len(work))
iwork := make([]int, m)
l := tmp.asTriDense(m, blas.NonUnit, blas.Lower)
v := lapack64.Trcon(lnorm, l.mat, work, iwork)
putWorkspace(tmp)
return 1 / v
}
// Cond returns the condition number of the given matrix under the given norm.
// The condition number must be based on the 1-norm, 2-norm or ∞-norm.
// Cond will panic with matrix.ErrShape if the matrix has zero size.
//
// BUG(btracey): The computation of the 1-norm and ∞-norm for non-square matrices
// is innacurate, although is typically the right order of magnitude. See
// https://github.com/xianyi/OpenBLAS/issues/636. While the value returned will
// change with the resolution of this bug, the result from Cond will match the
// condition number used internally.
func Cond(a Matrix, norm float64) float64 {
m, n := a.Dims()
if m == 0 || n == 0 {
panic(matrix.ErrShape)
}
var lnorm lapack.MatrixNorm
switch norm {
default:
panic("mat64: bad norm value")
case 1:
lnorm = lapack.MaxColumnSum
case 2:
// Use SVD to compute the condition number.
// TODO(btracey): Replace this with temporary workspace when SVD is fixed.
tmp := NewDense(m, n, nil)
tmp.Copy(a)
svd := SVD(tmp, math.Pow(2, -52.0), math.Pow(2, -966.0), false, false)
return svd.Cond()
case math.Inf(1):
lnorm = lapack.MaxRowSum
}
if m == n {
// Use the LU decomposition to compute the condition number.
tmp := getWorkspace(m, n, false)
tmp.Copy(a)
work := make([]float64, 4*n)
aNorm := lapack64.Lange(lnorm, tmp.mat, work)
pivot := make([]int, m)
lapack64.Getrf(tmp.mat, pivot)
iwork := make([]int, n)
v := lapack64.Gecon(lnorm, tmp.mat, aNorm, work, iwork)
putWorkspace(tmp)
return 1 / v
}
if m > n {
// Use the QR factorization to compute the condition number.
tmp := getWorkspace(m, n, false)
tmp.Copy(a)
work := make([]float64, 3*n)
tau := make([]float64, min(m, n))
lapack64.Geqrf(tmp.mat, tau, work, -1)
if int(work[0]) > len(work) {
work = make([]float64, int(work[0]))
}
lapack64.Geqrf(tmp.mat, tau, work, len(work))
iwork := make([]int, n)
r := tmp.asTriDense(n, blas.NonUnit, blas.Upper)
v := lapack64.Trcon(lnorm, r.mat, work, iwork)
putWorkspace(tmp)
return 1 / v
}
// Use the LQ factorization to compute the condition number.
tmp := getWorkspace(m, n, false)
tmp.Copy(a)
work := make([]float64, 3*m)
tau := make([]float64, min(m, n))
lapack64.Gelqf(tmp.mat, tau, work, -1)
if int(work[0]) > len(work) {
work = make([]float64, int(work[0]))
}
lapack64.Gelqf(tmp.mat, tau, work, len(work))
iwork := make([]int, m)
l := tmp.asTriDense(m, blas.NonUnit, blas.Lower)
v := lapack64.Trcon(lnorm, l.mat, work, iwork)
putWorkspace(tmp)
return 1 / v
}
