func (lq *LQ) updateCond() {
// A = LQ, where Q is orthonormal. Orthonormal multiplications do not change
// the condition number. Thus, ||A|| = ||L|| ||Q|| = ||Q||.
m := lq.lq.mat.Rows
work := make([]float64, 3*m)
iwork := make([]int, m)
l := lq.lq.asTriDense(m, blas.NonUnit, blas.Lower)
v := lapack64.Trcon(matrix.CondNorm, l.mat, work, iwork)
lq.cond = 1 / v
}
func (qr *QR) updateCond() {
// A = QR, where Q is orthonormal. Orthonormal multiplications do not change
// the condition number. Thus, ||A|| = ||Q|| ||R|| = ||R||.
n := qr.qr.mat.Cols
work := make([]float64, 3*n)
iwork := make([]int, n)
r := qr.qr.asTriDense(n, blas.NonUnit, blas.Upper)
v := lapack64.Trcon(matrix.CondNorm, r.mat, work, iwork)
qr.cond = 1 / v
}
// 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
}
// 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
}
// 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)
}
}