--- /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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