--- /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"
+
+// Dlasq4 computes an approximation to the smallest eigenvalue using values of d
+// from the previous transform.
+// i0, n0, and n0in are zero-indexed.
+//
+// Dlasq4 is an internal routine. It is exported for testing purposes.
+func (impl Implementation) Dlasq4(i0, n0 int, z []float64, pp int, n0in int, dmin, dmin1, dmin2, dn, dn1, dn2, tau float64, ttype int, g float64) (tauOut float64, ttypeOut int, gOut float64) {
+ const (
+ cnst1 = 0.563
+ cnst2 = 1.01
+ cnst3 = 1.05
+
+ cnstthird = 0.333 // TODO(btracey): Fix?
+ )
+ // A negative dmin forces the shift to take that absolute value
+ // ttype records the type of shift.
+ if dmin <= 0 {
+ tau = -dmin
+ ttype = -1
+ return tau, ttype, g
+ }
+ nn := 4*(n0+1) + pp - 1 // -1 for zero indexing
+ s := math.NaN() // Poison s so that failure to take a path below is obvious
+ if n0in == n0 {
+ // No eigenvalues deflated.
+ if dmin == dn || dmin == dn1 {
+ b1 := math.Sqrt(z[nn-3]) * math.Sqrt(z[nn-5])
+ b2 := math.Sqrt(z[nn-7]) * math.Sqrt(z[nn-9])
+ a2 := z[nn-7] + z[nn-5]
+ if dmin == dn && dmin1 == dn1 {
+ gap2 := dmin2 - a2 - dmin2/4
+ var gap1 float64
+ if gap2 > 0 && gap2 > b2 {
+ gap1 = a2 - dn - (b2/gap2)*b2
+ } else {
+ gap1 = a2 - dn - (b1 + b2)
+ }
+ if gap1 > 0 && gap1 > b1 {
+ s = math.Max(dn-(b1/gap1)*b1, 0.5*dmin)
+ ttype = -2
+ } else {
+ s = 0
+ if dn > b1 {
+ s = dn - b1
+ }
+ if a2 > b1+b2 {
+ s = math.Min(s, a2-(b1+b2))
+ }
+ s = math.Max(s, cnstthird*dmin)
+ ttype = -3
+ }
+ } else {
+ ttype = -4
+ s = dmin / 4
+ var gam float64
+ var np int
+ if dmin == dn {
+ gam = dn
+ a2 = 0
+ if z[nn-5] > z[nn-7] {
+ return tau, ttype, g
+ }
+ b2 = z[nn-5] / z[nn-7]
+ np = nn - 9
+ } else {
+ np = nn - 2*pp
+ gam = dn1
+ if z[np-4] > z[np-2] {
+ return tau, ttype, g
+ }
+ a2 = z[np-4] / z[np-2]
+ if z[nn-9] > z[nn-11] {
+ return tau, ttype, g
+ }
+ b2 = z[nn-9] / z[nn-11]
+ np = nn - 13
+ }
+ // Approximate contribution to norm squared from i < nn-1.
+ a2 += b2
+ for i4loop := np + 1; i4loop >= 4*(i0+1)-1+pp; i4loop -= 4 {
+ i4 := i4loop - 1
+ if b2 == 0 {
+ break
+ }
+ b1 = b2
+ if z[i4] > z[i4-2] {
+ return tau, ttype, g
+ }
+ b2 *= z[i4] / z[i4-2]
+ a2 += b2
+ if 100*math.Max(b2, b1) < a2 || cnst1 < a2 {
+ break
+ }
+ }
+ a2 *= cnst3
+ // Rayleigh quotient residual bound.
+ if a2 < cnst1 {
+ s = gam * (1 - math.Sqrt(a2)) / (1 + a2)
+ }
+ }
+ } else if dmin == dn2 {
+ ttype = -5
+ s = dmin / 4
+ // Compute contribution to norm squared from i > nn-2.
+ np := nn - 2*pp
+ b1 := z[np-2]
+ b2 := z[np-6]
+ gam := dn2
+ if z[np-8] > b2 || z[np-4] > b1 {
+ return tau, ttype, g
+ }
+ a2 := (z[np-8] / b2) * (1 + z[np-4]/b1)
+ // Approximate contribution to norm squared from i < nn-2.
+ if n0-i0 > 2 {
+ b2 = z[nn-13] / z[nn-15]
+ a2 += b2
+ for i4loop := (nn + 1) - 17; i4loop >= 4*(i0+1)-1+pp; i4loop -= 4 {
+ i4 := i4loop - 1
+ if b2 == 0 {
+ break
+ }
+ b1 = b2
+ if z[i4] > z[i4-2] {
+ return tau, ttype, g
+ }
+ b2 *= z[i4] / z[i4-2]
+ a2 += b2
+ if 100*math.Max(b2, b1) < a2 || cnst1 < a2 {
+ break
+ }
+ }
+ a2 *= cnst3
+ }
+ if a2 < cnst1 {
+ s = gam * (1 - math.Sqrt(a2)) / (1 + a2)
+ }
+ } else {
+ // Case 6, no information to guide us.
+ if ttype == -6 {
+ g += cnstthird * (1 - g)
+ } else if ttype == -18 {
+ g = cnstthird / 4
+ } else {
+ g = 1.0 / 4
+ }
+ s = g * dmin
+ ttype = -6
+ }
+ } else if n0in == (n0 + 1) {
+ // One eigenvalue just deflated. Use DMIN1, DN1 for DMIN and DN.
+ if dmin1 == dn1 && dmin2 == dn2 {
+ ttype = -7
+ s = cnstthird * dmin1
+ if z[nn-5] > z[nn-7] {
+ return tau, ttype, g
+ }
+ b1 := z[nn-5] / z[nn-7]
+ b2 := b1
+ if b2 != 0 {
+ for i4loop := 4*(n0+1) - 9 + pp; i4loop >= 4*(i0+1)-1+pp; i4loop -= 4 {
+ i4 := i4loop - 1
+ a2 := b1
+ if z[i4] > z[i4-2] {
+ return tau, ttype, g
+ }
+ b1 *= z[i4] / z[i4-2]
+ b2 += b1
+ if 100*math.Max(b1, a2) < b2 {
+ break
+ }
+ }
+ }
+ b2 = math.Sqrt(cnst3 * b2)
+ a2 := dmin1 / (1 + b2*b2)
+ gap2 := 0.5*dmin2 - a2
+ if gap2 > 0 && gap2 > b2*a2 {
+ s = math.Max(s, a2*(1-cnst2*a2*(b2/gap2)*b2))
+ } else {
+ s = math.Max(s, a2*(1-cnst2*b2))
+ ttype = -8
+ }
+ } else {
+ s = dmin1 / 4
+ if dmin1 == dn1 {
+ s = 0.5 * dmin1
+ }
+ ttype = -9
+ }
+ } else if n0in == (n0 + 2) {
+ // Two eigenvalues deflated. Use DMIN2, DN2 for DMIN and DN.
+ if dmin2 == dn2 && 2*z[nn-5] < z[nn-7] {
+ ttype = -10
+ s = cnstthird * dmin2
+ if z[nn-5] > z[nn-7] {
+ return tau, ttype, g
+ }
+ b1 := z[nn-5] / z[nn-7]
+ b2 := b1
+ if b2 != 0 {
+ for i4loop := 4*(n0+1) - 9 + pp; i4loop >= 4*(i0+1)-1+pp; i4loop -= 4 {
+ i4 := i4loop - 1
+ if z[i4] > z[i4-2] {
+ return tau, ttype, g
+ }
+ b1 *= z[i4] / z[i4-2]
+ b2 += b1
+ if 100*b1 < b2 {
+ break
+ }
+ }
+ }
+ b2 = math.Sqrt(cnst3 * b2)
+ a2 := dmin2 / (1 + b2*b2)
+ gap2 := z[nn-7] + z[nn-9] - math.Sqrt(z[nn-11])*math.Sqrt(z[nn-9]) - a2
+ if gap2 > 0 && gap2 > b2*a2 {
+ s = math.Max(s, a2*(1-cnst2*a2*(b2/gap2)*b2))
+ } else {
+ s = math.Max(s, a2*(1-cnst2*b2))
+ }
+ } else {
+ s = dmin2 / 4
+ ttype = -11
+ }
+ } else if n0in > n0+2 {
+ // Case 12, more than two eigenvalues deflated. No information.
+ s = 0
+ ttype = -12
+ }
+ tau = s
+ return tau, ttype, g
+}
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