dhseqr.go

// Copyright ©2016 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 testlapack

import (
	"fmt"
	"math"
	"testing"

	"gonum.org/v1/gonum/blas"
	"gonum.org/v1/gonum/blas/blas64"
	"gonum.org/v1/gonum/floats"
	"gonum.org/v1/gonum/lapack"
)

type Dhseqrer interface {
	Dhseqr(job lapack.SchurJob, compz lapack.SchurComp, n, ilo, ihi int, h []float64, ldh int, wr, wi []float64,
		z []float64, ldz int, work []float64, lwork int) int
}

type dhseqrTest struct {
	n   int
	ilo int
	ihi int
	h   []float64
	tol float64

	evWant []complex128
}

func DhseqrTest(t *testing.T, impl Dhseqrer) {
	for i, tc := range dhseqrTests {
		for _, job := range []lapack.SchurJob{lapack.EigenvaluesOnly, lapack.EigenvaluesAndSchur} {
			for _, wantz := range []bool{false, true} {
				for _, extra := range []int{0, 11} {
					testDhseqr(t, impl, i, tc, job, wantz, extra, true)
					testDhseqr(t, impl, i, tc, job, wantz, extra, false)
				}
			}
		}
	}
}

func testDhseqr(t *testing.T, impl Dhseqrer, i int, test dhseqrTest, job lapack.SchurJob, wantz bool, extra int, optwork bool) {
	const tol = 1e-14
	evTol := test.tol
	if evTol == 0 {
		evTol = tol
	}

	n := test.n
	ihi := test.ihi
	ilo := test.ilo
	h := zeros(n, n, n+extra)
	copyGeneral(h, blas64.General{Rows: n, Cols: n, Stride: max(1, n), Data: test.h})
	hCopy := cloneGeneral(h)

	compz := lapack.SchurNone
	z := blas64.General{Stride: max(1, n)}
	if wantz {
		// First, let Dhseqr initialize Z to the identity matrix.
		compz = lapack.SchurHess
		z = nanGeneral(n, n, n+extra)
	}

	wr := nanSlice(n)
	wi := nanSlice(n)

	work := nanSlice(max(1, n))
	if optwork {
		impl.Dhseqr(job, lapack.SchurHess, n, ilo, ihi, h.Data, h.Stride, wr, wi, z.Data, z.Stride, work, -1)
		work = nanSlice(int(work[0]))
	}

	unconverged := impl.Dhseqr(job, compz, n, ilo, ihi, h.Data, h.Stride, wr, wi, z.Data, z.Stride, work, len(work))
	prefix := fmt.Sprintf("Case %v: job=%c, compz=%c, n=%v, ilo=%v, ihi=%v, extra=%v, optwk=%v",
		i, job, compz, n, ilo, ihi, extra, optwork)
	if unconverged > 0 {
		t.Logf("%v: Dhseqr did not compute all eigenvalues. unconverged=%v", prefix, unconverged)
		if unconverged <= ilo {
			t.Fatalf("%v: 0 < unconverged <= ilo", prefix)
		}
	}

	// Check that wr and wi have been assigned completely.
	if floats.HasNaN(wr) {
		t.Errorf("%v: wr has NaN elements", prefix)
	}
	if floats.HasNaN(wi) {
		t.Errorf("%v: wi has NaN elements", prefix)
	}

	// Check that complex eigenvalues are stored in consecutive elements as
	// complex conjugate pairs.
	for i := 0; i < n; {
		if unconverged > 0 && i == ilo {
			// Skip the unconverged eigenvalues.
			i = unconverged
			continue
		}
		if wi[i] == 0 {
			// Real eigenvalue.
			i++
			continue
		}
		// Complex conjugate pair.
		if wr[i] != wr[i+1] {
			t.Errorf("%v: conjugate pair has real parts unequal", prefix)
		}
		if wi[i] < 0 {
			t.Errorf("%v: first in conjugate pair has negative imaginary part", prefix)
		}
		if wi[i+1] != -wi[i] {
			t.Errorf("%v: complex pair is not conjugate", prefix)
		}
		i += 2
	}

	// Check that H contains the Schur form T.
	if job == lapack.EigenvaluesAndSchur {
		for i := 0; i < n; {
			if unconverged > 0 && i == ilo {
				// Skip the unconverged eigenvalues.
				i = unconverged
				continue
			}
			if wi[i] == 0 {
				// Real eigenvalue.
				if wr[i] != h.Data[i*h.Stride+i] {
					t.Errorf("%v: T not in Schur form (real eigenvalue not on diagonal)", prefix)
				}
				i++
				continue
			}
			// Complex conjugate pair.
			im := math.Sqrt(math.Abs(h.Data[(i+1)*h.Stride+i])) * math.Sqrt(math.Abs(h.Data[i*h.Stride+i+1]))
			if wr[i] != h.Data[i*h.Stride+i] || wr[i] != h.Data[(i+1)*h.Stride+i+1] ||
				math.Abs(wi[i]-im) > tol {
				t.Errorf("%v: conjugate pair and 2×2 diagonal block don't correspond", prefix)
			}
			i += 2
		}
	}

	// Check that all the found eigenvalues are really eigenvalues.
	foundEV := make([]bool, len(test.evWant))
	for i := 0; i < n; {
		if unconverged > 0 && i == ilo {
			// Skip the unconverged eigenvalues.
			i = unconverged
			continue
		}
		ev := complex(wr[i], wi[i])
		// Use problem-specific tolerance for testing eigenvalues.
		found, index := containsComplex(test.evWant, ev, evTol)
		if !found {
			t.Errorf("%v: unexpected eigenvalue %v", prefix, ev)
		} else {
			foundEV[index] = true
		}
		i++
	}
	if unconverged == 0 {
		// Check that all eigenvalues have been found.
		// This simple check assumes that all eigenvalues are
		// sufficiently separated from each other at least by evTol.
		for i := range foundEV {
			if !foundEV[i] {
				t.Errorf("%v: %vth eigenvalue not found", prefix, i)
			}
		}
	}

	if !wantz {
		return
	}

	// Z must be orthogonal.
	if resid := residualOrthogonal(z, false); resid > tol*float64(n) {
		t.Errorf("Case %v: Z is not orthogonal; resid=%v, want<=%v", prefix, resid, tol*float64(n))
	}

	if job == lapack.EigenvaluesAndSchur {
		tz := zeros(n, n, n)
		blas64.Gemm(blas.NoTrans, blas.Trans, 1, h, z, 0, tz)
		ztz := zeros(n, n, n)
		blas64.Gemm(blas.NoTrans, blas.NoTrans, 1, z, tz, 0, ztz)
		if !equalApproxGeneral(ztz, hCopy, evTol) {
			t.Errorf("%v: H != Z T Zᵀ", prefix)
		}
	}

	// Restore H.
	copyGeneral(h, hCopy)
	// Call Dhseqr again with the identity matrix given explicitly in Q.
	q := eye(n, n+extra)
	impl.Dhseqr(job, lapack.SchurOrig, n, ilo, ihi, h.Data, h.Stride, wr, wi, q.Data, q.Stride, work, len(work))
	if !equalApproxGeneral(z, q, 0) {
		t.Errorf("%v: Z and Q are not equal", prefix)
	}
}

var dhseqrTests = []dhseqrTest{
	{
		n:   0,
		ilo: 0,
		ihi: -1,
	},
	{
		n:      1,
		ilo:    0,
		ihi:    0,
		h:      []float64{0},
		evWant: []complex128{0},
	},
	{
		n:      1,
		ilo:    0,
		ihi:    0,
		h:      []float64{7.09965484086874e-1},
		evWant: []complex128{7.09965484086874e-1},
	},
	{
		n:      2,
		ilo:    0,
		ihi:    1,
		h:      []float64{0, 0, 0, 0},
		evWant: []complex128{0},
	},
	{
		n:   2,
		ilo: 0,
		ihi: 1,
		h: []float64{
			1, 0,
			0, 1,
		},
		evWant: []complex128{1},
	},
	{
		n:   2,
		ilo: 0,
		ihi: 1,
		h: []float64{
			0, -1,
			1, 0,
		},
		evWant: []complex128{1i, -1i},
	},
	{
		n:   2,
		ilo: 0,
		ihi: 1,
		h: []float64{
			6.25219991450918e-1, 8.17510791994361e-1,
			3.31218891622294e-1, 1.24103744878131e-1,
		},
		evWant: []complex128{9.52203547663447e-1, -2.02879811334398e-1},
	},
	{
		n:   4,
		ilo: 1,
		ihi: 2,
		h: []float64{
			1, 0, 0, 0,
			0, 6.25219991450918e-1, 8.17510791994361e-1, 0,
			0, 3.31218891622294e-1, 1.24103744878131e-1, 0,
			0, 0, 0, 2,
		},
		evWant: []complex128{1, 2, 9.52203547663447e-1, -2.02879811334398e-1},
	},
	{
		n:   2,
		ilo: 0,
		ihi: 1,
		h: []float64{
			-1.1219562276608, 6.85473513349362e-1,
			-8.19951061145131e-1, 1.93728523178888e-1,
		},
		evWant: []complex128{
			-4.64113852240958e-1 + 3.59580510817350e-1i,
			-4.64113852240958e-1 - 3.59580510817350e-1i,
		},
	},
	{
		n:   5,
		ilo: 0,
		ihi: 4,
		h: []float64{
			9.57590178533658e-1, -5.10651295522708e-1, 9.24974510015869e-1, -1.30016306879522e-1, 2.92601986926954e-2,
			-1.08084756637964, 1.77529701001213, -1.36480197632509, 2.23196371219601e-1, 1.12912853063308e-1,
			0, -8.44075612174676e-1, 1.067867614486, -2.55782915176399e-1, -2.00598563137468e-1,
			0, 0, -5.67097237165410e-1, 2.07205057427341e-1, 6.54998340743380e-1,
			0, 0, 0, -1.89441413886041e-1, -4.18125416021786e-1,
		},
		evWant: []complex128{
			2.94393309555622,
			4.97029793606701e-1 + 3.63041654992384e-1i,
			4.97029793606701e-1 - 3.63041654992384e-1i,
			-1.74079119166145e-1 + 2.01570009462092e-1i,
			-1.74079119166145e-1 - 2.01570009462092e-1i,
		},
	},
	{
		// BFW62A matrix from MatrixMarket, balanced and factorized into
		// upper Hessenberg form in Octave.
		// Eigenvalues computed by eig function in Octave.
		// Dhseqr considers this matrix small (n <= 75).
		n:   62,
		ilo: 0,
		ihi: 61,
		tol: 1e-12,
		h: []float64{
			0.7610708, -0.71474042262732, -1.03373461417302e-17, 8.218284875369092e-18, -4.39003777724509e-18, -7.633870714681998e-18, -9.951525116511751e-18, -6.538760279193677e-18, -1.656240811786753e-18, -4.915424973452908e-18, -7.590492820502813e-18, -4.532592864746854e-18, 1.137360639223451e-18, -2.088794138001457e-18, 4.330727699351238e-18, 4.88172964159538e-18, -6.438459345602974e-18, 7.414402965763168e-18, 8.592387304092668e-19, 2.905505475188102e-18, -5.210204793418634e-18, 2.377023457149656e-18, -1.958364175388968e-18, -8.746122759061733e-20, 1.839015672758814e-18, 3.097454207400904e-18, 1.545856657360309e-18, 1.713923049773744e-21, 1.333951071201153e-18, -1.256151066318485e-18, 5.489655201308922e-19, -2.191335276195054e-18, 3.211054779957158e-18, -4.585099368362507e-19, 4.064807180521144e-18, -3.621561583390336e-18, 4.638803832189033e-19, 1.306132013406548e-18, 7.71710094138792e-19, 2.320760977517361e-18, -6.538298612520761e-19, -3.498839827985687e-18, 2.699116500378558e-18, -1.653627855476782e-18, 7.006984532830204e-19, -2.829108657299736e-18, -5.260772120044258e-18, 1.014346572590618e-18, -1.751563831849658e-18, -1.850186018112724e-19, 2.870415308417256e-18, -8.423700664162806e-19, 3.498345394735042e-19, 1.448350507022323e-18, -3.119800500343431e-18, 4.170966784863917e-18, -4.413795207992463e-19, -2.550853151356032e-18, 2.058575286932081e-18, 1.11609155804576e-18, -2.819648393130598e-18, -2.691434041700446e-19,
			-0.71474042262732, 1.938395745278447, -1.179041092366627, 2.994496337305918e-17, -1.886890458028042e-17, -1.696051150581149e-17, -1.066654178739982e-17, -6.29828959344471e-18, -3.017613522337327e-18, -6.087774183044001e-18, -1.623722797471356e-17, -1.609066043798334e-17, 1.1222991062801e-17, -1.199415856132727e-17, 3.254808477409891e-17, 3.032574771313234e-17, -2.950307698970009e-17, 3.390164991463985e-17, 3.580652213399182e-17, -1.363239170451582e-17, 5.318959619432346e-19, -2.094826370136215e-17, 2.503105756608372e-17, 1.02172885473953e-17, 2.072365201436764e-17, 5.419721687057681e-17, 2.946627912791419e-17, -1.348815353957071e-17, -6.109538294302727e-18, 8.919596400140564e-19, -4.965011212156456e-18, -8.892555328798407e-18, -8.533370652960768e-18, 2.670935841220357e-17, 6.294143735482237e-17, -6.631964574578866e-17, 1.610888675267686e-17, -1.914212496162892e-17, 7.491989417296875e-17, 2.433601037912241e-17, -3.471972221532949e-17, 1.239231453972595e-18, 8.477927675716922e-17, 1.832368873558869e-17, -3.15614722379294e-17, -3.481308523858143e-17, 8.753655535825511e-17, -2.311094913977048e-17, -1.97736043621337e-17, -4.591312368362259e-18, -4.090767028097941e-17, 9.630744841167777e-17, 6.376373378245906e-17, 3.517117740157962e-18, -4.504796150838874e-17, 7.517908631343054e-17, 5.08947280115584e-17, -1.382149857389238e-17, -2.198919507523352e-18, 7.280187019637593e-18, -4.195544216643798e-17, -1.69351463438629e-16,
			0, -1.179041092366627, 1.770797890903813, 0.9251612011433773, 0.003661446613681521, -0.005110479724841567, 0.0005014497289028411, 0.00268637752732762, -0.003137790817099964, 0.0001251340392028716, 0.005236816389348038, -0.007596537304410319, -0.003216883293048434, 0.002836879838039065, 0.006069520262676079, 0.002668518333518884, 0.009131523505676264, 0.004626632495050776, 0.00181579510454105, 0.001123900611304629, 0.001677485377614088, 0.00363754391761903, -0.0007470406844762735, 0.002833257933303097, -0.0002375815340930377, -0.002849024657372476, -0.0002629124003065383, -0.0009435052243060314, -0.001501677820905836, 0.0007867717317979819, -0.003255814847476796, 0.0001229596171032013, 0.001899497807037465, 0.001586135347885108, -0.002155598204409179, -0.004095921626627291, -0.004749259183184092, -0.0003901939289968589, -0.00126397970751315, 0.001155644134671306, 0.000291837164140963, -0.0008540793573406925, 0.0005498493646965147, -0.000512747881346778, 0.001308619245804509, -0.001057469234737898, 8.364932352209563e-05, -0.0004759538737373512, 0.0002832407173157385, -2.502295625364179e-05, -0.001116422235449543, 0.0008744887175767913, -0.001577030646119032, 0.0006630051771088335, 0.0008174386956312274, 0.0005885600522543514, -0.000529324123745168, 0.0006462360717703873, -0.0005292503950906053, -0.0008409757119361148, 0.001179292096398777, -7.042243897656571e-05,
			0, 0, 0.9256234889252499, 2.188661152704918, -1.116554836905615, -0.02243454666837822, 0.001818477623431723, 0.01031345522542573, -0.02255181751639313, 0.01064389372187288, 0.02644236613167033, -0.03711341885334873, -0.01368507023909662, 0.005876976997280983, 0.04065670610764593, 0.01666467627475975, 0.04720670725274503, 0.02074371568474985, 0.02492606640786733, -0.0007477579555839802, 0.008780018993769038, 0.02228545976823896, -0.001309713708470288, 0.02515283874752867, -0.002695437114195642, -0.03058898397162627, -0.004721936411831751, -0.01112709820777293, 0.005156744291122814, 0.001266520229863774, -0.0173310067274492, 0.006032177243961209, 0.01546906824241716, 0.004122912475005991, -0.01457685983271352, -0.02472315268907353, -0.0146012575743626, 0.01613627552485761, -0.03570849487210562, 0.009634392591632741, 0.01396210133689555, -0.01085301942984549, 0.02308071799914048, -0.01534806641683973, 0.007461896675297811, 0.001678793578211992, -0.004550709803423801, -0.003543952787451961, 0.01328644094056202, 0.01722933205866279, -0.01604068924890588, 0.01061524273934986, -0.01948663408394128, 0.02488657490054273, 0.002560204204893632, -0.007049942019567433, -0.005861737475764968, 0.007063905438215945, -0.01302170441839208, -0.01452116548010346, 0.008856002952171451, 0.003352799875293177,
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			0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.173908380119132, 1.450043378319349, 0.02765965035599782,
			0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.02984095753646376, 1.759629990218614,
		},
		evWant: []complex128{
			-0.01716884621227793,
			-0.1844331609734137,
			0.05200651487352193,
			0.1336851109127556,
			0.2020936631953806,
			0.3566470363060668,
			0.3627207699831158,
			0.4388555152488943,
			0.4776853636435153,
			0.5598821450074998,
			0.6249350549980947,
			0.6791310689291772,
			0.985877008147705 + 0.01929363300192029i,
			0.985877008147705 - 0.01929363300192029i,
			0.9908483217835712,
			1.011990761364073,
			1.130046345264462,
			1.323698071765709,
			1.348598229483672,
			1.363190626641638 + 0.05400660173350781i,
			1.363190626641638 - 0.05400660173350781i,
			1.632832316477256,
			1.646239548684271,
			1.742738908029794,
			1.763069014878969,
			1.789601126186348,
			1.945228042429256,
			1.946373262056993,
			1.997152389795002,
			2.261322781594966,
			2.286944100979884,
			2.447712649969029,
			2.557857506923634,
			2.608379034892128,
			2.653335615965549,
			2.675270309760236,
			2.964219802766918 + 0.01767482509567778i,
			2.964219802766918 - 0.01767482509567778i,
			3.014604817775139,
			3.158289371199042,
			3.311794215710077,
			3.389894197376214,
			3.553333074879877,
			3.641274442796651,
			3.857556223000175,
			4.045817381029103,
			4.330901939363569,
			4.337313647767939,
			4.527400487637496,
			4.917229128467305,
			4.985609414964091,
			5.687686849958608,
			5.79422309012183,
			5.997813119506478,
			6.732426637899077,
			6.957609338485607,
			7.529842664573327,
			7.609108287806763,
			7.761261355516287,
			8.311941758006698,
			9.07053741884884,
			9.217944588000314,
		},
	},
	{
		// TOLS90 matrix from MatrixMarket, balanced and factorized into
		// upper Hessenberg form in Octave.
		// Eigenvalues computed by eig function in Octave.
		// Dhseqr considers this matrix big (n > 75).
		n:   90,
		ilo: 0,
		ihi: 89,
		tol: 1e-12,
		h: []float64{
			0, 15.87097792952162, -0.08018778146407576, 1.923854288576926, 0.2308884192434597, 0.3394611053158814, -0.137843056509317, 0.08273319299962868, 0.09094485832017182, -0.1932984706702652, 0.1180502933247899, -0.2497600660136686, 0.05058238526630414, 0.1559958264692184, -0.1159258671363344, 0.009240657808439554, -0.07576925270220009, -0.08136563335139357, 0.03812188405626555, 0.08890970810264923, 0.03109600803532232, -0.01649595143487545, -0.03210884292964193, -0.06264912164116895, 0.02576130685189339, 0.07030469513274047, -0.03577429596420342, -0.06755584096587573, -0.01609449751643862, -0.034609250889499, -0.006417485131372289, -0.03097242984568944, 0.009636702832094541, 0.03557221482377193, 0.03197869740177413, 0.006433398911967483, -0.01135683279223704, 0.005555822920782794, 0.005082647305173025, 0.009385366191684759, 0.008128896608623586, -0.02296712755081439, -0.003777413820214802, 0.005123235117691023, -0.001849213959133196, -0.02340110883416673, 0.03713005378535863, 0.001732375396842803, -0.002750608017025007, -0.008067437745595896, 0.007330916864526544, -0.002616316716217023, -0.006297132815168344, -0.005559964495164493, -0.009996463242082429, 0.005115046921377182, 0.001491016327339477, -0.000957347151456719, 0.006745577006930539, -0.004544035710017501, -0.004411949528299773, -0.006799740101884459, 0.0005344071027975155, 0.0004690847920096184, -0.004718575501368472, -0.002540214169642457, 0.00357159293854215, -0.003314789809413553, -0.003402372051470523, 0.0006527769815598059, -0.002716170091878954, -0.001540321517093821, 0.001756589868588563, -0.001379260966541515, 2.681259838616508e-05, -0.004307760317855304, -0.001109087032949234, -0.0001659937040461595, -0.0002235436314482387, -0.00119569899683125, 0.0005623327348935116, -0.0006320070189143994, 0.001043573892153891, -0.0008750118344396425, 0.0005348270042878935, 0.0008273143853381587, -0.0006622544067919339, -0.0008492128694001077, -0.0007986665099336194, -0.0009015882205421384,
			-12.61872336344661, -0.7232497785660141, 7.609159226285555, 0.3870426982100038, 5.924856640096555, 0.6102682886559223, -0.1693535996869482, -1.919018099340693, -0.2138609560373459, 0.8042623392399718, 0.8674124147325857, 1.331963567449537, -1.782290696158441, -0.8346237530841022, -1.552156412468403, -0.6151273144224881, -0.7391936693539152, 0.9311934770410842, 0.6166024938842418, -1.54644771695033, -1.776481997993627, -0.934164018650274, 1.955449870926906, -2.082936149197492, -0.222464503236819, 0.1331142802435182, -0.7633944660137448, -0.8176194198678164, 0.2746498608375185, -0.1119720547510174, -0.3046093297897633, 0.093708523663493, -0.1254384253653835, -0.3826156782538976, 0.9720657328245648, 0.4435611080311743, -0.002705519800257373, 0.4068502033494004, 0.8228026417475635, 1.898141183528268, -2.19419418629088, -2.26250233043472, 1.169369389476936, -2.320900757809857, 1.963641620262248, -0.9292511308866737, 1.20566753941007, -3.909359844030778, -1.745106869690986, -0.1674293744185332, 1.824325508201922, -0.611032518780414, -0.8172751651339325, -0.379367811436249, -1.118039132172379, -2.1156899896767, 0.6317312221182597, 1.675601832555065, -0.1958681401344015, 3.202902011360347, 0.2083144054490269, 1.651454396883902, -1.879950006164014, 1.114962429097019, -3.728120039552464, 2.115800549383747, 1.993994934513307, 1.036800346894651, 2.843012388647532, 2.665528597975866, -0.08443708961414848, -1.658265513220315, 2.825691629937317, 1.163068598831327, 1.032612155987785, -0.3464359914213002, -0.7374155713124153, -0.009825846309687306, -1.878310233217088, 1.038183449333641, 0.8683442149070821, 0.2096953654957043, 1.097533010763627, -1.40566207568186, 1.539718090250373, -0.3057191046244732, 0.9596910019319377, 0.4410690813618931, -0.01734977865916698, -0.7028527205475527,
			0, -21.14096588949502, -11.77071611766029, 91.38817973790069, 24.01994594814793, 7.139034668143506, -7.77684013545977, 11.8189112903465, 4.350214240774185, -13.41161300072837, 0.8429515510200203, -0.1960485971162483, -1.892239827255886, -2.481138864200454, -4.993335484535367, -1.462743953230215, -1.649744938703439, -3.306123402194819, 0.6299802354863298, 0.006312776908129815, 0.7029217322720615, 0.1969979096896496, 0.6484394805159026, 0.4612739081461551, 1.218623710778157, 1.937283920899595, -4.274393204137636, -1.417028232311232, 1.761975679896983, -1.207846701077455, -2.947009378633724, -1.111456928119195, 2.001483835367479, -0.3273985739357373, 1.27142153234808, 2.470518283440578, -0.497592748085565, -1.01776837453108, 0.1736631347448449, 0.2170284795451557, -1.102797734602059, -0.8983239378242805, 0.8376082516437703, -1.015605628895311, 1.888662040615523, -1.813946159105028, 3.745871277192266, -1.691058864813766, 0.5544744103796291, -0.95962769863539, 1.495178118153111, 0.1369976805376806, -3.134133785033962, -1.134766199832475, -2.142472928604951, 0.4360359463007911, 1.080773790484485, 0.9136687613609559, 1.421868877932455, -0.4939788885367615, 0.01579620756278606, -0.4041188740514539, -0.3370957888349073, 0.4032844546374829, -1.281049156842126, 1.334866305935072, -0.4288161314791398, -0.4076960827986253, 0.8975402184760907, 0.008483305617251051, -0.02300021991545602, -0.4779200297205075, 0.8640559542693361, 0.6740166979504152, 1.271173988705413, -0.125268363712917, -0.02783785192682384, -0.7952831181002197, -0.1834134462936435, -0.3033906534395994, -0.1842466257842833, 0.3294577790658342, 0.4265858101126394, -1.050551059782219, 0.5557053448408287, -0.07549099980078718, 0.4485085972651929, 0.604772326452039, 0.02668915155289681, -0.35958437456048,
			0, 0, -71.67241414253803, -12.6925239065222, -58.51026881403752, -9.923972440030834, -18.08337737101079, 23.73437311293314, 10.42624729020936, -7.39628437264571, 6.93334510855383, -16.20186028661058, 19.09098951338167, 7.231158491837533, 25.86051183536314, 4.692849852643402, 13.50562601772541, -23.8959783279092, 10.46795736800158, 26.42552249365325, 25.55147405605497, 30.98455252329583, -38.26426057779729, 41.72459242758727, 14.01187370572939, -9.227025126956866, 17.69341876066779, 17.7058942448478, -13.2182156659012, 3.273277215940703, 7.82096407961303, -6.540134857551783, 5.302726399754003, 21.6168229759513, -35.45773268931614, -39.5771763658349, 16.91907564224343, 16.55810765769962, 10.84170890675409, 4.539182245882841, 5.966139065523301, 10.43624249585104, -6.306944364017387, 25.11309378553394, -23.98463112536712, -3.253554932408899, -26.20327820737005, 34.95346757431584, -22.06848426488626, 0.3787905650745691, 12.79793983153946, -15.15662916810503, 30.10820985389998, 17.18239253486946, 26.44238280127885, -17.53916820777098, -21.59984930800645, -20.89069843604591, -10.62614963929214, -4.16861219455282, -15.8281632392319, -15.45413424684084, 7.28126264287692, -17.30978445489622, 25.83878482866339, -18.20409699627451, 1.341279973912325, 6.060701390094514, -0.9930555858249585, -0.4770067839263625, -8.050723322366107, 15.00633993006825, -10.12301020599794, -15.02267685265745, -28.87546819977945, -4.887384695963816, 3.812851132430744, 19.10682316350844, 7.478580657962908, -7.486104775378393, 8.24517564743228, -7.51905558097683, -6.644004835040043, 16.04319982377311, -12.24015636297471, -0.4152718733052826, -15.37260980717165, -14.36579932723399, -1.30513875795791, 5.356894027365968,
			0, 0, 0, 88.73441744471468, -9.667604704172753, -22.84943872004879, 5.044285909895449, -2.76004875036668, 0.3050715430090338, 2.281066203460998, -3.081685045157843, 5.563173468318147, 1.196920824152984, 0.7508125732458136, 3.261490552895722, -0.1915889730517104, 2.178042983750966, -5.719722606964433, -0.808717831824074, 3.567750315780947, 4.405620166473985, 8.068713509838501, -3.974147763943233, 4.024558504483837, 1.343369353638846, -1.554164148226279, 1.397410693322202, 1.549071510171349, -0.6861822032616559, -0.03144220974090741, -1.503375319686877, -1.213563509509453, 1.485658175240746, 1.102628808733887, -2.836353822887371, -2.651198571403894, 1.944770277514162, 1.800720538730825, 2.925044559436877, 2.489394773406274, -3.99962439548742, -1.365153890760246, 1.729186611640552, 0.90732580595284, 0.02970965138899272, -9.802670261550649, -3.377885239589956, 3.311681996737421, -2.102890043550365, -2.752571666784161, 7.725320034635699, -0.8099349955071451, 0.5680350525364315, 2.461090824551101, 1.671002962178604, -4.651796111745832, -3.019627268376409, -1.456848301794733, 0.6765376227892419, -2.208873565004413, -2.755468168868538, -2.723938546851782, 1.844529360647301, -1.962817053181678, 2.77164872022158, -0.6312645341808628, 0.9959732581947718, -0.01141276076056162, -1.047045912982139, 0.9659655272370172, -1.069371622104567, -0.564318267120407, -0.9606489624515147, -0.3731646049074267, -0.4361324066751411, -1.228953081985217, 2.112718903093324, 3.774673462225595, 2.038526286226191, -2.037134562294253, -0.6272498215703869, -0.9130826952549337, -0.5356753186936942, 1.739354952287502, -2.607676632661601, 1.167128690015535, -3.643945638175477, -2.247788469333459, 0.4447578257714688, 0.8814188227164721,
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		},
		evWant: []complex128{
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			-0.2513649571095319 - 26.5196181977274i,
			-0.3323427794627154 + 14.12147023570828i,
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			-1.641962795944021 - 124.0727533257146i,
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			-24.04314341198735 + 0.3315626206724418i,
			-24.04314341198735 - 0.3315626206724418i,
			-24.12753399862331 + 0.1634093642552133i,
			-24.12753399862331 - 0.1634093642552133i,
			-24.18103712080912,
			-24.18908889661277,
			-24.19357072143492,
			-24.19567681157636,
			-24.20660003986661,
			-24.228086198612 + 0.0220672004906673i,
			-24.228086198612 - 0.0220672004906673i,
			-24.30677145425284 + 0.1813999700804981i,
			-24.30677145425284 - 0.1813999700804981i,
			-25.23862373866253 + 0.8733279005402655i,
			-25.23862373866253 - 0.8733279005402655i,
			-29.42427841595277 + 1.666886570068163i,
			-29.42427841595277 - 1.666886570068163i,
			-33.94959271558832,
			-35.51958918286584,
			-36.20666045708104 + 0.1660921702962803i,
			-36.20666045708104 - 0.1660921702962803i,
			-36.25710351845593 + 0.01052251476620707i,
			-36.25710351845593 - 0.01052251476620707i,
			-36.28714605125749,
			-36.29431583135849,
			-36.29711199023137,
			-36.31304183793699,
			-36.32232768169418,
			-36.37871120972082 + 0.1557647672076243i,
			-36.37871120972082 - 0.1557647672076243i,
			-36.47694233727385 + 0.3738707259908102i,
			-36.47694233727385 - 0.3738707259908102i,
			-36.66483183150206,
			-37.75937189360096,
		},
	},
	{
		// TUB100 matrix from MatrixMarket, balanced and factorized into
		// upper Hessenberg form in Octave, and embedded into a 104×104
		// diagonal matrix to test with ilo != 0 and ihi != n-1.
		// Eigenvalues computed by eig function in Octave.
		// Dhseqr considers this matrix big (n > 75).
		n:   104,
		ilo: 2,
		ihi: 101,
		tol: 1e-10,
		h: []float64{
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