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 // 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) { switch { case i0 < 0: panic(i0LT0) case n0 < 0: panic(n0LT0) case len(z) < 4*n0: panic(shortZ) case pp != 0 && pp != 1: panic(badPp) } 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 }