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// Copyright ©2016 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package gonum
import (
"math"
"gonum.org/v1/gonum/blas/blas64"
"gonum.org/v1/gonum/lapack"
)
// Dgebal balances an n×n matrix A. Balancing consists of two stages, permuting
// and scaling. Both steps are optional and depend on the value of job.
//
// Permuting consists of applying a permutation matrix P such that the matrix
// that results from Pᵀ*A*P takes the upper block triangular form
// [ T1 X Y ]
// Pᵀ A P = [ 0 B Z ],
// [ 0 0 T2 ]
// where T1 and T2 are upper triangular matrices and B contains at least one
// nonzero off-diagonal element in each row and column. The indices ilo and ihi
// mark the starting and ending columns of the submatrix B. The eigenvalues of A
// isolated in the first 0 to ilo-1 and last ihi+1 to n-1 elements on the
// diagonal can be read off without any roundoff error.
//
// Scaling consists of applying a diagonal similarity transformation D such that
// D^{-1}*B*D has the 1-norm of each row and its corresponding column nearly
// equal. The output matrix is
// [ T1 X*D Y ]
// [ 0 inv(D)*B*D inv(D)*Z ].
// [ 0 0 T2 ]
// Scaling may reduce the 1-norm of the matrix, and improve the accuracy of
// the computed eigenvalues and/or eigenvectors.
//
// job specifies the operations that will be performed on A.
// If job is lapack.BalanceNone, Dgebal sets scale[i] = 1 for all i and returns ilo=0, ihi=n-1.
// If job is lapack.Permute, only permuting will be done.
// If job is lapack.Scale, only scaling will be done.
// If job is lapack.PermuteScale, both permuting and scaling will be done.
//
// On return, if job is lapack.Permute or lapack.PermuteScale, it will hold that
// A[i,j] == 0, for i > j and j ∈ {0, ..., ilo-1, ihi+1, ..., n-1}.
// If job is lapack.BalanceNone or lapack.Scale, or if n == 0, it will hold that
// ilo == 0 and ihi == n-1.
//
// On return, scale will contain information about the permutations and scaling
// factors applied to A. If π(j) denotes the index of the column interchanged
// with column j, and D[j,j] denotes the scaling factor applied to column j,
// then
// scale[j] == π(j), for j ∈ {0, ..., ilo-1, ihi+1, ..., n-1},
// == D[j,j], for j ∈ {ilo, ..., ihi}.
// scale must have length equal to n, otherwise Dgebal will panic.
//
// Dgebal is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dgebal(job lapack.BalanceJob, n int, a []float64, lda int, scale []float64) (ilo, ihi int) {
switch {
case job != lapack.BalanceNone && job != lapack.Permute && job != lapack.Scale && job != lapack.PermuteScale:
panic(badBalanceJob)
case n < 0:
panic(nLT0)
case lda < max(1, n):
panic(badLdA)
}
ilo = 0
ihi = n - 1
if n == 0 {
return ilo, ihi
}
if len(scale) != n {
panic(shortScale)
}
if job == lapack.BalanceNone {
for i := range scale {
scale[i] = 1
}
return ilo, ihi
}
if len(a) < (n-1)*lda+n {
panic(shortA)
}
bi := blas64.Implementation()
swapped := true
if job == lapack.Scale {
goto scaling
}
// Permutation to isolate eigenvalues if possible.
//
// Search for rows isolating an eigenvalue and push them down.
for swapped {
swapped = false
rows:
for i := ihi; i >= 0; i-- {
for j := 0; j <= ihi; j++ {
if i == j {
continue
}
if a[i*lda+j] != 0 {
continue rows
}
}
// Row i has only zero off-diagonal elements in the
// block A[ilo:ihi+1,ilo:ihi+1].
scale[ihi] = float64(i)
if i != ihi {
bi.Dswap(ihi+1, a[i:], lda, a[ihi:], lda)
bi.Dswap(n, a[i*lda:], 1, a[ihi*lda:], 1)
}
if ihi == 0 {
scale[0] = 1
return ilo, ihi
}
ihi--
swapped = true
break
}
}
// Search for columns isolating an eigenvalue and push them left.
swapped = true
for swapped {
swapped = false
columns:
for j := ilo; j <= ihi; j++ {
for i := ilo; i <= ihi; i++ {
if i == j {
continue
}
if a[i*lda+j] != 0 {
continue columns
}
}
// Column j has only zero off-diagonal elements in the
// block A[ilo:ihi+1,ilo:ihi+1].
scale[ilo] = float64(j)
if j != ilo {
bi.Dswap(ihi+1, a[j:], lda, a[ilo:], lda)
bi.Dswap(n-ilo, a[j*lda+ilo:], 1, a[ilo*lda+ilo:], 1)
}
swapped = true
ilo++
break
}
}
scaling:
for i := ilo; i <= ihi; i++ {
scale[i] = 1
}
if job == lapack.Permute {
return ilo, ihi
}
// Balance the submatrix in rows ilo to ihi.
const (
// sclfac should be a power of 2 to avoid roundoff errors.
// Elements of scale are restricted to powers of sclfac,
// therefore the matrix will be only nearly balanced.
sclfac = 2
// factor determines the minimum reduction of the row and column
// norms that is considered non-negligible. It must be less than 1.
factor = 0.95
)
sfmin1 := dlamchS / dlamchP
sfmax1 := 1 / sfmin1
sfmin2 := sfmin1 * sclfac
sfmax2 := 1 / sfmin2
// Iterative loop for norm reduction.
var conv bool
for !conv {
conv = true
for i := ilo; i <= ihi; i++ {
c := bi.Dnrm2(ihi-ilo+1, a[ilo*lda+i:], lda)
r := bi.Dnrm2(ihi-ilo+1, a[i*lda+ilo:], 1)
ica := bi.Idamax(ihi+1, a[i:], lda)
ca := math.Abs(a[ica*lda+i])
ira := bi.Idamax(n-ilo, a[i*lda+ilo:], 1)
ra := math.Abs(a[i*lda+ilo+ira])
// Guard against zero c or r due to underflow.
if c == 0 || r == 0 {
continue
}
g := r / sclfac
f := 1.0
s := c + r
for c < g && math.Max(f, math.Max(c, ca)) < sfmax2 && math.Min(r, math.Min(g, ra)) > sfmin2 {
if math.IsNaN(c + f + ca + r + g + ra) {
// Panic if NaN to avoid infinite loop.
panic("lapack: NaN")
}
f *= sclfac
c *= sclfac
ca *= sclfac
g /= sclfac
r /= sclfac
ra /= sclfac
}
g = c / sclfac
for r <= g && math.Max(r, ra) < sfmax2 && math.Min(math.Min(f, c), math.Min(g, ca)) > sfmin2 {
f /= sclfac
c /= sclfac
ca /= sclfac
g /= sclfac
r *= sclfac
ra *= sclfac
}
if c+r >= factor*s {
// Reduction would be negligible.
continue
}
if f < 1 && scale[i] < 1 && f*scale[i] <= sfmin1 {
continue
}
if f > 1 && scale[i] > 1 && scale[i] >= sfmax1/f {
continue
}
// Now balance.
scale[i] *= f
bi.Dscal(n-ilo, 1/f, a[i*lda+ilo:], 1)
bi.Dscal(ihi+1, f, a[i:], lda)
conv = false
}
}
return ilo, ihi
}