+++ /dev/null
-// 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) {
- // TODO(btracey): make info an error.
- if len(z) < 4*n {
- panic(badZ)
- }
- const cbias = 1.5
-
- eps := dlamchP
- safmin := dlamchS
- tol := eps * 100
- tol2 := tol * tol
- if n < 0 {
- panic(nLT0)
- }
- if n == 0 {
- return info
- }
- if n == 1 {
- if z[0] < 0 {
- panic(negZ)
- }
- return info
- }
- if n == 2 {
- if z[1] < 0 || z[2] < 0 {
- panic("lapack: bad z value")
- } 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("lapack: bad z value")
- }
- 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("lapack: bad z value")
- }
- d += z[2*(n-1)]
- qmax = math.Max(qmax, 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
- n1 = n0
- 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
-}
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