// Factorize computes the LQ factorization of an m×n matrix a where n <= m. The LQ
// factorization always exists even if A is singular.
//
// The LQ decomposition is a factorization of the matrix A such that A = L * Q.
// The matrix Q is an orthonormal n×n matrix, and L is an m×n upper triangular matrix.
// L and Q can be extracted from the LFromLQ and QFromLQ methods on Dense.
func (lq *LQ) Factorize(a Matrix) {
m, n := a.Dims()
if m > n {
panic(ErrShape)
}
k := min(m, n)
if lq.lq == nil {
lq.lq = &Dense{}
}
lq.lq.Clone(a)
work := make([]float64, 1)
lq.tau = make([]float64, k)
lapack64.Gelqf(lq.lq.mat, lq.tau, work, -1)
work = make([]float64, int(work[0]))
lapack64.Gelqf(lq.lq.mat, lq.tau, work, len(work))
}
// 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
}