blob: fd24a5509a4250c47edc7352efc35f79d2d65bec [file] [log] [blame]
// Copyright ©2015 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/lapack"
)
// Dlasq2 computes all the eigenvalues of the symmetric positive
// definite tridiagonal matrix associated with the qd array Z. Eigevalues
// are computed to high relative accuracy avoiding denormalization, underflow
// and overflow.
//
// To see the relation of Z to the tridiagonal matrix, let L be a
// unit lower bidiagonal matrix with sub-diagonals Z(2,4,6,,..) and
// let U be an upper bidiagonal matrix with 1's above and diagonal
// Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the
// symmetric tridiagonal to which it is similar.
//
// info returns a status error. The return codes mean as follows:
// 0: The algorithm completed successfully.
// 1: A split was marked by a positive value in e.
// 2: Current block of Z not diagonalized after 100*n iterations (in inner
// while loop). On exit Z holds a qd array with the same eigenvalues as
// the given Z.
// 3: Termination criterion of outer while loop not met (program created more
// than N unreduced blocks).
//
// z must have length at least 4*n, and must not contain any negative elements.
// Dlasq2 will panic otherwise.
//
// Dlasq2 is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dlasq2(n int, z []float64) (info int) {
if n < 0 {
panic(nLT0)
}
if n == 0 {
return info
}
if len(z) < 4*n {
panic(shortZ)
}
if n == 1 {
if z[0] < 0 {
panic(negZ)
}
return info
}
const cbias = 1.5
eps := dlamchP
safmin := dlamchS
tol := eps * 100
tol2 := tol * tol
if n == 2 {
if z[1] < 0 || z[2] < 0 {
panic(negZ)
} else if z[2] > z[0] {
z[0], z[2] = z[2], z[0]
}
z[4] = z[0] + z[1] + z[2]
if z[1] > z[2]*tol2 {
t := 0.5 * (z[0] - z[2] + z[1])
s := z[2] * (z[1] / t)
if s <= t {
s = z[2] * (z[1] / (t * (1 + math.Sqrt(1+s/t))))
} else {
s = z[2] * (z[1] / (t + math.Sqrt(t)*math.Sqrt(t+s)))
}
t = z[0] + s + z[1]
z[2] *= z[0] / t
z[0] = t
}
z[1] = z[2]
z[5] = z[1] + z[0]
return info
}
// Check for negative data and compute sums of q's and e's.
z[2*n-1] = 0
emin := z[1]
var d, e, qmax float64
var i1, n1 int
for k := 0; k < 2*(n-1); k += 2 {
if z[k] < 0 || z[k+1] < 0 {
panic(negZ)
}
d += z[k]
e += z[k+1]
qmax = math.Max(qmax, z[k])
emin = math.Min(emin, z[k+1])
}
if z[2*(n-1)] < 0 {
panic(negZ)
}
d += z[2*(n-1)]
// Check for diagonality.
if e == 0 {
for k := 1; k < n; k++ {
z[k] = z[2*k]
}
impl.Dlasrt(lapack.SortDecreasing, n, z)
z[2*(n-1)] = d
return info
}
trace := d + e
// Check for zero data.
if trace == 0 {
z[2*(n-1)] = 0
return info
}
// Rearrange data for locality: Z=(q1,qq1,e1,ee1,q2,qq2,e2,ee2,...).
for k := 2 * n; k >= 2; k -= 2 {
z[2*k-1] = 0
z[2*k-2] = z[k-1]
z[2*k-3] = 0
z[2*k-4] = z[k-2]
}
i0 := 0
n0 := n - 1
// Reverse the qd-array, if warranted.
// z[4*i0-3] --> z[4*(i0+1)-3-1] --> z[4*i0]
if cbias*z[4*i0] < z[4*n0] {
ipn4Out := 4 * (i0 + n0 + 2)
for i4loop := 4 * (i0 + 1); i4loop <= 2*(i0+n0+1); i4loop += 4 {
i4 := i4loop - 1
ipn4 := ipn4Out - 1
z[i4-3], z[ipn4-i4-4] = z[ipn4-i4-4], z[i4-3]
z[i4-1], z[ipn4-i4-6] = z[ipn4-i4-6], z[i4-1]
}
}
// Initial split checking via dqd and Li's test.
pp := 0
for k := 0; k < 2; k++ {
d = z[4*n0+pp]
for i4loop := 4*n0 + pp; i4loop >= 4*(i0+1)+pp; i4loop -= 4 {
i4 := i4loop - 1
if z[i4-1] <= tol2*d {
z[i4-1] = math.Copysign(0, -1)
d = z[i4-3]
} else {
d = z[i4-3] * (d / (d + z[i4-1]))
}
}
// dqd maps Z to ZZ plus Li's test.
emin = z[4*(i0+1)+pp]
d = z[4*i0+pp]
for i4loop := 4*(i0+1) + pp; i4loop <= 4*n0+pp; i4loop += 4 {
i4 := i4loop - 1
z[i4-2*pp-2] = d + z[i4-1]
if z[i4-1] <= tol2*d {
z[i4-1] = math.Copysign(0, -1)
z[i4-2*pp-2] = d
z[i4-2*pp] = 0
d = z[i4+1]
} else if safmin*z[i4+1] < z[i4-2*pp-2] && safmin*z[i4-2*pp-2] < z[i4+1] {
tmp := z[i4+1] / z[i4-2*pp-2]
z[i4-2*pp] = z[i4-1] * tmp
d *= tmp
} else {
z[i4-2*pp] = z[i4+1] * (z[i4-1] / z[i4-2*pp-2])
d = z[i4+1] * (d / z[i4-2*pp-2])
}
emin = math.Min(emin, z[i4-2*pp])
}
z[4*(n0+1)-pp-3] = d
// Now find qmax.
qmax = z[4*(i0+1)-pp-3]
for i4loop := 4*(i0+1) - pp + 2; i4loop <= 4*(n0+1)+pp-2; i4loop += 4 {
i4 := i4loop - 1
qmax = math.Max(qmax, z[i4])
}
// Prepare for the next iteration on K.
pp = 1 - pp
}
// Initialise variables to pass to DLASQ3.
var ttype int
var dmin1, dmin2, dn, dn1, dn2, g, tau float64
var tempq float64
iter := 2
var nFail int
nDiv := 2 * (n0 - i0)
var i4 int
outer:
for iwhila := 1; iwhila <= n+1; iwhila++ {
// Test for completion.
if n0 < 0 {
// Move q's to the front.
for k := 1; k < n; k++ {
z[k] = z[4*k]
}
// Sort and compute sum of eigenvalues.
impl.Dlasrt(lapack.SortDecreasing, n, z)
e = 0
for k := n - 1; k >= 0; k-- {
e += z[k]
}
// Store trace, sum(eigenvalues) and information on performance.
z[2*n] = trace
z[2*n+1] = e
z[2*n+2] = float64(iter)
z[2*n+3] = float64(nDiv) / float64(n*n)
z[2*n+4] = 100 * float64(nFail) / float64(iter)
return info
}
// While array unfinished do
// e[n0] holds the value of sigma when submatrix in i0:n0
// splits from the rest of the array, but is negated.
var desig float64
var sigma float64
if n0 != n-1 {
sigma = -z[4*(n0+1)-2]
}
if sigma < 0 {
info = 1
return info
}
// Find last unreduced submatrix's top index i0, find qmax and
// emin. Find Gershgorin-type bound if Q's much greater than E's.
var emax float64
if n0 > i0 {
emin = math.Abs(z[4*(n0+1)-6])
} else {
emin = 0
}
qmin := z[4*(n0+1)-4]
qmax = qmin
zSmall := false
for i4loop := 4 * (n0 + 1); i4loop >= 8; i4loop -= 4 {
i4 = i4loop - 1
if z[i4-5] <= 0 {
zSmall = true
break
}
if qmin >= 4*emax {
qmin = math.Min(qmin, z[i4-3])
emax = math.Max(emax, z[i4-5])
}
qmax = math.Max(qmax, z[i4-7]+z[i4-5])
emin = math.Min(emin, z[i4-5])
}
if !zSmall {
i4 = 3
}
i0 = (i4+1)/4 - 1
pp = 0
if n0-i0 > 1 {
dee := z[4*i0]
deemin := dee
kmin := i0
for i4loop := 4*(i0+1) + 1; i4loop <= 4*(n0+1)-3; i4loop += 4 {
i4 := i4loop - 1
dee = z[i4] * (dee / (dee + z[i4-2]))
if dee <= deemin {
deemin = dee
kmin = (i4+4)/4 - 1
}
}
if (kmin-i0)*2 < n0-kmin && deemin <= 0.5*z[4*n0] {
ipn4Out := 4 * (i0 + n0 + 2)
pp = 2
for i4loop := 4 * (i0 + 1); i4loop <= 2*(i0+n0+1); i4loop += 4 {
i4 := i4loop - 1
ipn4 := ipn4Out - 1
z[i4-3], z[ipn4-i4-4] = z[ipn4-i4-4], z[i4-3]
z[i4-2], z[ipn4-i4-3] = z[ipn4-i4-3], z[i4-2]
z[i4-1], z[ipn4-i4-6] = z[ipn4-i4-6], z[i4-1]
z[i4], z[ipn4-i4-5] = z[ipn4-i4-5], z[i4]
}
}
}
// Put -(initial shift) into DMIN.
dmin := -math.Max(0, qmin-2*math.Sqrt(qmin)*math.Sqrt(emax))
// Now i0:n0 is unreduced.
// PP = 0 for ping, PP = 1 for pong.
// PP = 2 indicates that flipping was applied to the Z array and
// and that the tests for deflation upon entry in Dlasq3
// should not be performed.
nbig := 100 * (n0 - i0 + 1)
for iwhilb := 0; iwhilb < nbig; iwhilb++ {
if i0 > n0 {
continue outer
}
// While submatrix unfinished take a good dqds step.
i0, n0, pp, dmin, sigma, desig, qmax, nFail, iter, nDiv, ttype, dmin1, dmin2, dn, dn1, dn2, g, tau =
impl.Dlasq3(i0, n0, z, pp, dmin, sigma, desig, qmax, nFail, iter, nDiv, ttype, dmin1, dmin2, dn, dn1, dn2, g, tau)
pp = 1 - pp
// When emin is very small check for splits.
if pp == 0 && n0-i0 >= 3 {
if z[4*(n0+1)-1] <= tol2*qmax || z[4*(n0+1)-2] <= tol2*sigma {
splt := i0 - 1
qmax = z[4*i0]
emin = z[4*(i0+1)-2]
oldemn := z[4*(i0+1)-1]
for i4loop := 4 * (i0 + 1); i4loop <= 4*(n0-2); i4loop += 4 {
i4 := i4loop - 1
if z[i4] <= tol2*z[i4-3] || z[i4-1] <= tol2*sigma {
z[i4-1] = -sigma
splt = i4 / 4
qmax = 0
emin = z[i4+3]
oldemn = z[i4+4]
} else {
qmax = math.Max(qmax, z[i4+1])
emin = math.Min(emin, z[i4-1])
oldemn = math.Min(oldemn, z[i4])
}
}
z[4*(n0+1)-2] = emin
z[4*(n0+1)-1] = oldemn
i0 = splt + 1
}
}
}
// Maximum number of iterations exceeded, restore the shift
// sigma and place the new d's and e's in a qd array.
// This might need to be done for several blocks.
info = 2
i1 = i0
for {
tempq = z[4*i0]
z[4*i0] += sigma
for k := i0 + 1; k <= n0; k++ {
tempe := z[4*(k+1)-6]
z[4*(k+1)-6] *= tempq / z[4*(k+1)-8]
tempq = z[4*k]
z[4*k] += sigma + tempe - z[4*(k+1)-6]
}
// Prepare to do this on the previous block if there is one.
if i1 <= 0 {
break
}
n1 = i1 - 1
for i1 >= 1 && z[4*(i1+1)-6] >= 0 {
i1 -= 1
}
sigma = -z[4*(n1+1)-2]
}
for k := 0; k < n; k++ {
z[2*k] = z[4*k]
// Only the block 1..N0 is unfinished. The rest of the e's
// must be essentially zero, although sometimes other data
// has been stored in them.
if k < n0 {
z[2*(k+1)-1] = z[4*(k+1)-1]
} else {
z[2*(k+1)] = 0
}
}
return info
}
info = 3
return info
}