package("polmat")

"Library Polmat successfully loaded."

p:=(z+2)*(1/z+2)

(1/z + 2)*(z + 2)

polmat::isPos(p,[z],2^3)

TRUE

polmat::isPos(p,[z],2^2,all)

TRUE, array(1..4, 1 = 9, 2 = 5, 3 = 1, 4 = 5)

polmat::isPos(p,[z],2^3,all)

TRUE, array(1..8, 1 = 9, 2 = 2*2^(1/2) + 5, 3 = 5, 4 = 5 - 2*2^(1/2), 5 = 1, 6 = 5 - 2*2^(1/2), 7 = 5, 8 = 2*2^(1/2) + 5)

polmat::isPos(p,[z],2^3,Numeric,all)

TRUE, array(1..8, 1 = 9.0, 2 = 7.828427125 + 4.440892099*10^(-16)*I, 3 = 5.0, 4 = 2.171572875 + (- 6.172747013*10^(-16)*I), 5 = 1.0, 6 = 2.171572875 + (- 4.440892099*10^(-16)*I), 7 = 5.0, 8 = 7.828427125 + 6.172747013*10^(-16)*I)

p:=(z+2)*(1/z+2)+(v+4)*(1/v+4)*(w+1)*(1/w+1)

(1/z + 2)*(z + 2) + (1/v + 4)*(1/w + 1)*(v + 4)*(w + 1)

polmat::isPos(p,[z,v,w],2^3,Numeric,all)

      array(1..8, 1..8, 1..8,
        (1, 1, 1) = 109.0,
        (1, 1, 2) = 94.35533906 + 7.105427358e-15 I,
        (1, 1, 3) = 59.0,
        (1, 1, 4) = 23.64466094 - 5.717532321e-15 I,
        (1, 1, 5) = 9.0,
        (1, 1, 6) = 23.64466094 - 7.105427358e-15 I,
        (1, 1, 7) = 59.0,
        (1, 1, 8) = 94.35533906 + 5.717532321e-15 I,
        (1, 2, 1) = 99.627417 + 3.552713679e-15 I,
        (1, 2, 2) = 86.35533906 + 8.881784197e-15 I,
        (1, 2, 3) = 54.3137085 + 1.776356839e-15 I,
        (1, 2, 4) = 22.27207794 - 7.290989737e-15 I,
        (1, 2, 5) = 9.0,
        (1, 2, 6) = 22.27207794 - 5.329070518e-15 I,
        (1, 2, 7) = 54.3137085 + 1.776356839e-15 I,
        (1, 2, 8) = 86.35533906 + 1.084370342e-14 I,
        (1, 3, 1) = 77.0,
        (1, 3, 2) = 67.04163056 + 3.552713679e-15 I,
        (1, 3, 3) = 43.0,
        (1, 3, 4) = 18.95836944 - 5.024790356e-15 I,
        (1, 3, 5) = 9.0,
        (1, 3, 6) = 18.95836944 - 3.552713679e-15 I,
        (1, 3, 7) = 43.0,
        (1, 3, 8) = 67.04163056 + 5.024790356e-15 I,
        (1, 4, 1) = 54.372583 - 4.93819761e-15 I,
        (1, 4, 2) = 47.72792206 + 1.083614874e-15 I,
        (1, 4, 3) = 31.6862915 - 2.469098805e-15 I,
        (1, 4, 4) = 15.64466094 - 7.004046619e-15 I,
        (1, 4, 5) = 9.0,
        (1, 4, 6) = 15.64466094 - 6.021812484e-15 I,
        (1, 4, 7) = 31.6862915 - 2.469098805e-15 I,
        (1, 4, 8) = 47.72792206 + 2.065849009e-15 I,
        (1, 5, 1) = 45.0,
        (1, 5, 2) = 39.72792206 + 1.776356839e-15 I,
        (1, 5, 3) = 27.0,
        (1, 5, 4) = 14.27207794 - 3.44386997e-15 I,
        (1, 5, 5) = 9.0,
        (1, 5, 6) = 14.27207794 - 1.776356839e-15 I,
        (1, 5, 7) = 27.0,
        (1, 5, 8) = 39.72792206 + 3.44386997e-15 I,
        (1, 6, 1) = 54.372583 - 3.552713679e-15 I,
        (1, 6, 2) = 47.72792206 + 1.776356839e-15 I,
        (1, 6, 3) = 31.6862915 - 1.776356839e-15 I,
        (1, 6, 4) = 15.64466094 - 6.311304654e-15 I,
        (1, 6, 5) = 9.0,
        (1, 6, 6) = 15.64466094 - 5.329070518e-15 I,
        (1, 6, 7) = 31.6862915 - 1.776356839e-15 I,
        (1, 6, 8) = 47.72792206 + 2.758590975e-15 I,
        (1, 7, 1) = 77.0,
        (1, 7, 2) = 67.04163056 + 3.552713679e-15 I,
        (1, 7, 3) = 43.0,
        (1, 7, 4) = 18.95836944 - 5.024790356e-15 I,
        (1, 7, 5) = 9.0,
        (1, 7, 6) = 18.95836944 - 3.552713679e-15 I,
        (1, 7, 7) = 43.0,
        (1, 7, 8) = 67.04163056 + 5.024790356e-15 I,
        (1, 8, 1) = 99.627417 + 4.93819761e-15 I,
        (1, 8, 2) = 86.35533906 + 9.574526163e-15 I,
        (1, 8, 3) = 54.3137085 + 2.469098805e-15 I,
        (1, 8, 4) = 22.27207794 - 6.598247771e-15 I,
        (1, 8, 5) = 9.0,
        (1, 8, 6) = 22.27207794 - 4.636328553e-15 I,
        (1, 8, 7) = 54.3137085 + 2.469098805e-15 I,
        (1, 8, 8) = 86.35533906 + 1.153644538e-14 I,
        (2, 1, 1) = 107.8284271 + 4.440892099e-16 I,
        (2, 1, 2) = 93.18376618 + 7.549516567e-15 I,
        (2, 1, 3) = 57.82842712 + 4.440892099e-16 I,
        (2, 1, 4) = 22.47308807 - 5.273443112e-15 I,
        (2, 1, 5) = 7.828427125 + 4.440892099e-16 I,
        (2, 1, 6) = 22.47308807 - 6.661338148e-15 I,
        (2, 1, 7) = 57.82842712 + 4.440892099e-16 I,
        (2, 1, 8) = 93.18376618 + 6.161621531e-15 I,
        (2, 2, 1) = 98.45584412 + 3.996802889e-15 I,
        (2, 2, 2) = 85.18376618 + 9.325873407e-15 I,
        (2, 2, 3) = 53.14213562 + 2.220446049e-15 I,
        (2, 2, 4) = 21.10050506 - 6.846900527e-15 I,
        (2, 2, 5) = 7.828427125 + 4.440892099e-16 I,
        (2, 2, 6) = 21.10050506 - 4.884981308e-15 I,
        (2, 2, 7) = 53.14213562 + 2.220446049e-15 I,
        (2, 2, 8) = 85.18376618 + 1.128779263e-14 I,
        (2, 3, 1) = 75.82842712 + 4.440892099e-16 I,
        (2, 3, 2) = 65.87005769 + 3.996802889e-15 I,
        (2, 3, 3) = 41.82842712 + 4.440892099e-16 I,
        (2, 3, 4) = 17.78679656 - 4.580701146e-15 I,
        (2, 3, 5) = 7.828427125 + 4.440892099e-16 I,
        (2, 3, 6) = 17.78679656 - 3.108624469e-15 I,
        (2, 3, 7) = 41.82842712 + 4.440892099e-16 I,
        (2, 3, 8) = 65.87005769 + 5.468879566e-15 I,
        (2, 4, 1) = 53.20101013 - 4.4941084e-15 I,
        (2, 4, 2) = 46.55634919 + 1.527704084e-15 I,
        (2, 4, 3) = 30.51471863 - 2.025009595e-15 I,
        (2, 4, 4) = 14.47308807 - 6.55995741e-15 I,
        (2, 4, 5) = 7.828427125 + 4.440892099e-16 I,
        (2, 4, 6) = 14.47308807 - 5.577723274e-15 I,
        (2, 4, 7) = 30.51471863 - 2.025009595e-15 I,
        (2, 4, 8) = 46.55634919 + 2.509938219e-15 I,
        (2, 5, 1) = 43.82842712 + 4.440892099e-16 I,
        (2, 5, 2) = 38.55634919 + 2.220446049e-15 I,
        (2, 5, 3) = 25.82842712 + 4.440892099e-16 I,
        (2, 5, 4) = 13.10050506 - 2.999780761e-15 I,
        (2, 5, 5) = 7.828427125 + 4.440892099e-16 I,
        (2, 5, 6) = 13.10050506 - 1.33226763e-15 I,
        (2, 5, 7) = 25.82842712 + 4.440892099e-16 I,
        (2, 5, 8) = 38.55634919 + 3.88795918e-15 I,
        (2, 6, 1) = 53.20101013 - 3.108624469e-15 I,
        (2, 6, 2) = 46.55634919 + 2.220446049e-15 I,
        (2, 6, 3) = 30.51471863 - 1.33226763e-15 I,
        (2, 6, 4) = 14.47308807 - 5.867215444e-15 I,
        (2, 6, 5) = 7.828427125 + 4.440892099e-16 I,
        (2, 6, 6) = 14.47308807 - 4.884981308e-15 I,
        (2, 6, 7) = 30.51471863 - 1.33226763e-15 I,
        (2, 6, 8) = 46.55634919 + 3.202680185e-15 I,
        (2, 7, 1) = 75.82842712 + 4.440892099e-16 I,
        (2, 7, 2) = 65.87005769 + 3.996802889e-15 I,
        (2, 7, 3) = 41.82842712 + 4.440892099e-16 I,
        (2, 7, 4) = 17.78679656 - 4.580701146e-15 I,
        (2, 7, 5) = 7.828427125 + 4.440892099e-16 I,
        (2, 7, 6) = 17.78679656 - 3.108624469e-15 I,
        (2, 7, 7) = 41.82842712 + 4.440892099e-16 I,
        (2, 7, 8) = 65.87005769 + 5.468879566e-15 I,
        (2, 8, 1) = 98.45584412 + 5.38228682e-15 I,
        (2, 8, 2) = 85.18376618 + 1.001861537e-14 I,
        (2, 8, 3) = 53.14213562 + 2.913188015e-15 I,
        (2, 8, 4) = 21.10050506 - 6.154158561e-15 I,
        (2, 8, 5) = 7.828427125 + 4.440892099e-16 I,
        (2, 8, 6) = 21.10050506 - 4.192239343e-15 I,
        (2, 8, 7) = 53.14213562 + 2.913188015e-15 I,
        (2, 8, 8) = 85.18376618 + 1.198053459e-14 I,
        (3, 1, 1) = 105.0,
        (3, 1, 2) = 90.35533906 + 7.105427358e-15 I,
        (3, 1, 3) = 55.0,
        (3, 1, 4) = 19.64466094 - 5.717532321e-15 I,
        (3, 1, 5) = 5.0,
        (3, 1, 6) = 19.64466094 - 7.105427358e-15 I,
        (3, 1, 7) = 55.0,
        (3, 1, 8) = 90.35533906 + 5.717532321e-15 I,
        (3, 2, 1) = 95.627417 + 3.552713679e-15 I,
        (3, 2, 2) = 82.35533906 + 8.881784197e-15 I,
        (3, 2, 3) = 50.3137085 + 1.776356839e-15 I,
        (3, 2, 4) = 18.27207794 - 7.290989737e-15 I,
        (3, 2, 5) = 5.0,
        (3, 2, 6) = 18.27207794 - 5.329070518e-15 I,
        (3, 2, 7) = 50.3137085 + 1.776356839e-15 I,
        (3, 2, 8) = 82.35533906 + 1.084370342e-14 I,
        (3, 3, 1) = 73.0,
        (3, 3, 2) = 63.04163056 + 3.552713679e-15 I,
        (3, 3, 3) = 39.0,
        (3, 3, 4) = 14.95836944 - 5.024790356e-15 I,
        (3, 3, 5) = 5.0,
        (3, 3, 6) = 14.95836944 - 3.552713679e-15 I,
        (3, 3, 7) = 39.0,
        (3, 3, 8) = 63.04163056 + 5.024790356e-15 I,
        (3, 4, 1) = 50.372583 - 4.93819761e-15 I,
        (3, 4, 2) = 43.72792206 + 1.083614874e-15 I,
        (3, 4, 3) = 27.6862915 - 2.469098805e-15 I,
        (3, 4, 4) = 11.64466094 - 7.004046619e-15 I,
        (3, 4, 5) = 5.0,
        (3, 4, 6) = 11.64466094 - 6.021812484e-15 I,
        (3, 4, 7) = 27.6862915 - 2.469098805e-15 I,
        (3, 4, 8) = 43.72792206 + 2.065849009e-15 I,
        (3, 5, 1) = 41.0,
        (3, 5, 2) = 35.72792206 + 1.776356839e-15 I,
        (3, 5, 3) = 23.0,
        (3, 5, 4) = 10.27207794 - 3.44386997e-15 I,
        (3, 5, 5) = 5.0,
        (3, 5, 6) = 10.27207794 - 1.776356839e-15 I,
        (3, 5, 7) = 23.0,
        (3, 5, 8) = 35.72792206 + 3.44386997e-15 I,
        (3, 6, 1) = 50.372583 - 3.552713679e-15 I,
        (3, 6, 2) = 43.72792206 + 1.776356839e-15 I,
        (3, 6, 3) = 27.6862915 - 1.776356839e-15 I,
        (3, 6, 4) = 11.64466094 - 6.311304654e-15 I,
        (3, 6, 5) = 5.0,
        (3, 6, 6) = 11.64466094 - 5.329070518e-15 I,
        (3, 6, 7) = 27.6862915 - 1.776356839e-15 I,
        (3, 6, 8) = 43.72792206 + 2.758590975e-15 I,
        (3, 7, 1) = 73.0,
        (3, 7, 2) = 63.04163056 + 3.552713679e-15 I,
        (3, 7, 3) = 39.0,
        (3, 7, 4) = 14.95836944 - 5.024790356e-15 I,
        (3, 7, 5) = 5.0,
        (3, 7, 6) = 14.95836944 - 3.552713679e-15 I,
        (3, 7, 7) = 39.0,
        (3, 7, 8) = 63.04163056 + 5.024790356e-15 I,
        (3, 8, 1) = 95.627417 + 4.93819761e-15 I,
        (3, 8, 2) = 82.35533906 + 9.574526163e-15 I,
        (3, 8, 3) = 50.3137085 + 2.469098805e-15 I,
        (3, 8, 4) = 18.27207794 - 6.598247771e-15 I,
        (3, 8, 5) = 5.0,
        (3, 8, 6) = 18.27207794 - 4.636328553e-15 I,
        (3, 8, 7) = 50.3137085 + 2.469098805e-15 I,
        (3, 8, 8) = 82.35533906 + 1.153644538e-14 I,
        (4, 1, 1) = 102.1715729 - 6.172747013e-16 I,
        (4, 1, 2) = 87.52691193 + 6.488152656e-15 I,
        (4, 1, 3) = 52.17157288 - 6.172747013e-16 I,
        (4, 1, 4) = 16.81623382 - 6.334807023e-15 I,
        (4, 1, 5) = 2.171572875 - 6.172747013e-16 I,
        (4, 1, 6) = 16.81623382 - 7.722702059e-15 I,
        (4, 1, 7) = 52.17157288 - 6.172747013e-16 I,
        (4, 1, 8) = 87.52691193 + 5.10025762e-15 I,
        (4, 2, 1) = 92.79898987 + 2.935438978e-15 I,
        (4, 2, 2) = 79.52691193 + 8.264509496e-15 I,
        (4, 2, 3) = 47.48528137 + 1.159082138e-15 I,
        (4, 2, 4) = 15.44365081 - 7.908264438e-15 I,
        (4, 2, 5) = 2.171572875 - 6.172747013e-16 I,
        (4, 2, 6) = 15.44365081 - 5.946345219e-15 I,
        (4, 2, 7) = 47.48528137 + 1.159082138e-15 I,
        (4, 2, 8) = 79.52691193 + 1.022642871e-14 I,
        (4, 3, 1) = 70.17157288 - 6.172747013e-16 I,
        (4, 3, 2) = 60.21320344 + 2.935438978e-15 I,
        (4, 3, 3) = 36.17157288 - 6.172747013e-16 I,
        (4, 3, 4) = 12.12994231 - 5.642065057e-15 I,
        (4, 3, 5) = 2.171572875 - 6.172747013e-16 I,
        (4, 3, 6) = 12.12994231 - 4.16998838e-15 I,
        (4, 3, 7) = 36.17157288 - 6.172747013e-16 I,
        (4, 3, 8) = 60.21320344 + 4.407515655e-15 I,
        (4, 4, 1) = 47.54415588 - 5.555472311e-15 I,
        (4, 4, 2) = 40.89949494 + 4.663401725e-16 I,
        (4, 4, 3) = 24.85786438 - 3.086373506e-15 I,
        (4, 4, 4) = 8.816233816 - 7.621321321e-15 I,
        (4, 4, 5) = 2.171572875 - 6.172747013e-16 I,
        (4, 4, 6) = 8.816233816 - 6.639087185e-15 I,
        (4, 4, 7) = 24.85786438 - 3.086373506e-15 I,
        (4, 4, 8) = 40.89949494 + 1.448574308e-15 I,
        (4, 5, 1) = 38.17157288 - 6.172747013e-16 I,
        (4, 5, 2) = 32.89949494 + 1.159082138e-15 I,
        (4, 5, 3) = 20.17157288 - 6.172747013e-16 I,
        (4, 5, 4) = 7.443650814 - 4.061144672e-15 I,
        (4, 5, 5) = 2.171572875 - 6.172747013e-16 I,
        (4, 5, 6) = 7.443650814 - 2.393631541e-15 I,
        (4, 5, 7) = 20.17157288 - 6.172747013e-16 I,
        (4, 5, 8) = 32.89949494 + 2.826595269e-15 I,
        (4, 6, 1) = 47.54415588 - 4.16998838e-15 I,
        (4, 6, 2) = 40.89949494 + 1.159082138e-15 I,
        (4, 6, 3) = 24.85786438 - 2.393631541e-15 I,
        (4, 6, 4) = 8.816233816 - 6.928579355e-15 I,
        (4, 6, 5) = 2.171572875 - 6.172747013e-16 I,
        (4, 6, 6) = 8.816233816 - 5.946345219e-15 I,
        (4, 6, 7) = 24.85786438 - 2.393631541e-15 I,
        (4, 6, 8) = 40.89949494 + 2.141316274e-15 I,
        (4, 7, 1) = 70.17157288 - 6.172747013e-16 I,
        (4, 7, 2) = 60.21320344 + 2.935438978e-15 I,
        (4, 7, 3) = 36.17157288 - 6.172747013e-16 I,
        (4, 7, 4) = 12.12994231 - 5.642065057e-15 I,
        (4, 7, 5) = 2.171572875 - 6.172747013e-16 I,
        (4, 7, 6) = 12.12994231 - 4.16998838e-15 I,
        (4, 7, 7) = 36.17157288 - 6.172747013e-16 I,
        (4, 7, 8) = 60.21320344 + 4.407515655e-15 I,
        (4, 8, 1) = 92.79898987 + 4.320922909e-15 I,
        (4, 8, 2) = 79.52691193 + 8.957251461e-15 I,
        (4, 8, 3) = 47.48528137 + 1.851824104e-15 I,
        (4, 8, 4) = 15.44365081 - 7.215522472e-15 I,
        (4, 8, 5) = 2.171572875 - 6.172747013e-16 I,
        (4, 8, 6) = 15.44365081 - 5.253603254e-15 I,
        (4, 8, 7) = 47.48528137 + 1.851824104e-15 I,
TRUE,   (4, 8, 8) = 79.52691193 + 1.091917068e-14 I,
        (5, 1, 1) = 101.0,
        (5, 1, 2) = 86.35533906 + 7.105427358e-15 I,
        (5, 1, 3) = 51.0,
        (5, 1, 4) = 15.64466094 - 5.717532321e-15 I,
        (5, 1, 5) = 1.0,
        (5, 1, 6) = 15.64466094 - 7.105427358e-15 I,
        (5, 1, 7) = 51.0,
        (5, 1, 8) = 86.35533906 + 5.717532321e-15 I,
        (5, 2, 1) = 91.627417 + 3.552713679e-15 I,
        (5, 2, 2) = 78.35533906 + 8.881784197e-15 I,
        (5, 2, 3) = 46.3137085 + 1.776356839e-15 I,
        (5, 2, 4) = 14.27207794 - 7.290989737e-15 I,
        (5, 2, 5) = 1.0,
        (5, 2, 6) = 14.27207794 - 5.329070518e-15 I,
        (5, 2, 7) = 46.3137085 + 1.776356839e-15 I,
        (5, 2, 8) = 78.35533906 + 1.084370342e-14 I,
        (5, 3, 1) = 69.0,
        (5, 3, 2) = 59.04163056 + 3.552713679e-15 I,
        (5, 3, 3) = 35.0,
        (5, 3, 4) = 10.95836944 - 5.024790356e-15 I,
        (5, 3, 5) = 1.0,
        (5, 3, 6) = 10.95836944 - 3.552713679e-15 I,
        (5, 3, 7) = 35.0,
        (5, 3, 8) = 59.04163056 + 5.024790356e-15 I,
        (5, 4, 1) = 46.372583 - 4.93819761e-15 I,
        (5, 4, 2) = 39.72792206 + 1.083614874e-15 I,
        (5, 4, 3) = 23.6862915 - 2.469098805e-15 I,
        (5, 4, 4) = 7.644660941 - 7.004046619e-15 I,
        (5, 4, 5) = 1.0,
        (5, 4, 6) = 7.644660941 - 6.021812484e-15 I,
        (5, 4, 7) = 23.6862915 - 2.469098805e-15 I,
        (5, 4, 8) = 39.72792206 + 2.065849009e-15 I,
        (5, 5, 1) = 37.0,
        (5, 5, 2) = 31.72792206 + 1.776356839e-15 I,
        (5, 5, 3) = 19.0,
        (5, 5, 4) = 6.272077939 - 3.44386997e-15 I,
        (5, 5, 5) = 1.0,
        (5, 5, 6) = 6.272077939 - 1.776356839e-15 I,
        (5, 5, 7) = 19.0,
        (5, 5, 8) = 31.72792206 + 3.44386997e-15 I,
        (5, 6, 1) = 46.372583 - 3.552713679e-15 I,
        (5, 6, 2) = 39.72792206 + 1.776356839e-15 I,
        (5, 6, 3) = 23.6862915 - 1.776356839e-15 I,
        (5, 6, 4) = 7.644660941 - 6.311304654e-15 I,
        (5, 6, 5) = 1.0,
        (5, 6, 6) = 7.644660941 - 5.329070518e-15 I,
        (5, 6, 7) = 23.6862915 - 1.776356839e-15 I,
        (5, 6, 8) = 39.72792206 + 2.758590975e-15 I,
        (5, 7, 1) = 69.0,
        (5, 7, 2) = 59.04163056 + 3.552713679e-15 I,
        (5, 7, 3) = 35.0,
        (5, 7, 4) = 10.95836944 - 5.024790356e-15 I,
        (5, 7, 5) = 1.0,
        (5, 7, 6) = 10.95836944 - 3.552713679e-15 I,
        (5, 7, 7) = 35.0,
        (5, 7, 8) = 59.04163056 + 5.024790356e-15 I,
        (5, 8, 1) = 91.627417 + 4.93819761e-15 I,
        (5, 8, 2) = 78.35533906 + 9.574526163e-15 I,
        (5, 8, 3) = 46.3137085 + 2.469098805e-15 I,
        (5, 8, 4) = 14.27207794 - 6.598247771e-15 I,
        (5, 8, 5) = 1.0,
        (5, 8, 6) = 14.27207794 - 4.636328553e-15 I,
        (5, 8, 7) = 46.3137085 + 2.469098805e-15 I,
        (5, 8, 8) = 78.35533906 + 1.153644538e-14 I,
        (6, 1, 1) = 102.1715729 - 4.440892099e-16 I,
        (6, 1, 2) = 87.52691193 + 6.661338148e-15 I,
        (6, 1, 3) = 52.17157288 - 4.440892099e-16 I,
        (6, 1, 4) = 16.81623382 - 6.161621531e-15 I,
        (6, 1, 5) = 2.171572875 - 4.440892099e-16 I,
        (6, 1, 6) = 16.81623382 - 7.549516567e-15 I,
        (6, 1, 7) = 52.17157288 - 4.440892099e-16 I,
        (6, 1, 8) = 87.52691193 + 5.273443112e-15 I,
        (6, 2, 1) = 92.79898987 + 3.108624469e-15 I,
        (6, 2, 2) = 79.52691193 + 8.437694987e-15 I,
        (6, 2, 3) = 47.48528137 + 1.33226763e-15 I,
        (6, 2, 4) = 15.44365081 - 7.735078947e-15 I,
        (6, 2, 5) = 2.171572875 - 4.440892099e-16 I,
        (6, 2, 6) = 15.44365081 - 5.773159728e-15 I,
        (6, 2, 7) = 47.48528137 + 1.33226763e-15 I,
        (6, 2, 8) = 79.52691193 + 1.039961421e-14 I,
        (6, 3, 1) = 70.17157288 - 4.440892099e-16 I,
        (6, 3, 2) = 60.21320344 + 3.108624469e-15 I,
        (6, 3, 3) = 36.17157288 - 4.440892099e-16 I,
        (6, 3, 4) = 12.12994231 - 5.468879566e-15 I,
        (6, 3, 5) = 2.171572875 - 4.440892099e-16 I,
        (6, 3, 6) = 12.12994231 - 3.996802889e-15 I,
        (6, 3, 7) = 36.17157288 - 4.440892099e-16 I,
        (6, 3, 8) = 60.21320344 + 4.580701146e-15 I,
        (6, 4, 1) = 47.54415588 - 5.38228682e-15 I,
        (6, 4, 2) = 40.89949494 + 6.395256639e-16 I,
        (6, 4, 3) = 24.85786438 - 2.913188015e-15 I,
        (6, 4, 4) = 8.816233816 - 7.448135829e-15 I,
        (6, 4, 5) = 2.171572875 - 4.440892099e-16 I,
        (6, 4, 6) = 8.816233816 - 6.465901694e-15 I,
        (6, 4, 7) = 24.85786438 - 2.913188015e-15 I,
        (6, 4, 8) = 40.89949494 + 1.621759799e-15 I,
        (6, 5, 1) = 38.17157288 - 4.440892099e-16 I,
        (6, 5, 2) = 32.89949494 + 1.33226763e-15 I,
        (6, 5, 3) = 20.17157288 - 4.440892099e-16 I,
        (6, 5, 4) = 7.443650814 - 3.88795918e-15 I,
        (6, 5, 5) = 2.171572875 - 4.440892099e-16 I,
        (6, 5, 6) = 7.443650814 - 2.220446049e-15 I,
        (6, 5, 7) = 20.17157288 - 4.440892099e-16 I,
        (6, 5, 8) = 32.89949494 + 2.999780761e-15 I,
        (6, 6, 1) = 47.54415588 - 3.996802889e-15 I,
        (6, 6, 2) = 40.89949494 + 1.33226763e-15 I,
        (6, 6, 3) = 24.85786438 - 2.220446049e-15 I,
        (6, 6, 4) = 8.816233816 - 6.755393864e-15 I,
        (6, 6, 5) = 2.171572875 - 4.440892099e-16 I,
        (6, 6, 6) = 8.816233816 - 5.773159728e-15 I,
        (6, 6, 7) = 24.85786438 - 2.220446049e-15 I,
        (6, 6, 8) = 40.89949494 + 2.314501765e-15 I,
        (6, 7, 1) = 70.17157288 - 4.440892099e-16 I,
        (6, 7, 2) = 60.21320344 + 3.108624469e-15 I,
        (6, 7, 3) = 36.17157288 - 4.440892099e-16 I,
        (6, 7, 4) = 12.12994231 - 5.468879566e-15 I,
        (6, 7, 5) = 2.171572875 - 4.440892099e-16 I,
        (6, 7, 6) = 12.12994231 - 3.996802889e-15 I,
        (6, 7, 7) = 36.17157288 - 4.440892099e-16 I,
        (6, 7, 8) = 60.21320344 + 4.580701146e-15 I,
        (6, 8, 1) = 92.79898987 + 4.4941084e-15 I,
        (6, 8, 2) = 79.52691193 + 9.130436953e-15 I,
        (6, 8, 3) = 47.48528137 + 2.025009595e-15 I,
        (6, 8, 4) = 15.44365081 - 7.042336981e-15 I,
        (6, 8, 5) = 2.171572875 - 4.440892099e-16 I,
        (6, 8, 6) = 15.44365081 - 5.080417762e-15 I,
        (6, 8, 7) = 47.48528137 + 2.025009595e-15 I,
        (6, 8, 8) = 79.52691193 + 1.109235617e-14 I,
        (7, 1, 1) = 105.0,
        (7, 1, 2) = 90.35533906 + 7.105427358e-15 I,
        (7, 1, 3) = 55.0,
        (7, 1, 4) = 19.64466094 - 5.717532321e-15 I,
        (7, 1, 5) = 5.0,
        (7, 1, 6) = 19.64466094 - 7.105427358e-15 I,
        (7, 1, 7) = 55.0,
        (7, 1, 8) = 90.35533906 + 5.717532321e-15 I,
        (7, 2, 1) = 95.627417 + 3.552713679e-15 I,
        (7, 2, 2) = 82.35533906 + 8.881784197e-15 I,
        (7, 2, 3) = 50.3137085 + 1.776356839e-15 I,
        (7, 2, 4) = 18.27207794 - 7.290989737e-15 I,
        (7, 2, 5) = 5.0,
        (7, 2, 6) = 18.27207794 - 5.329070518e-15 I,
        (7, 2, 7) = 50.3137085 + 1.776356839e-15 I,
        (7, 2, 8) = 82.35533906 + 1.084370342e-14 I,
        (7, 3, 1) = 73.0,
        (7, 3, 2) = 63.04163056 + 3.552713679e-15 I,
        (7, 3, 3) = 39.0,
        (7, 3, 4) = 14.95836944 - 5.024790356e-15 I,
        (7, 3, 5) = 5.0,
        (7, 3, 6) = 14.95836944 - 3.552713679e-15 I,
        (7, 3, 7) = 39.0,
        (7, 3, 8) = 63.04163056 + 5.024790356e-15 I,
        (7, 4, 1) = 50.372583 - 4.93819761e-15 I,
        (7, 4, 2) = 43.72792206 + 1.083614874e-15 I,
        (7, 4, 3) = 27.6862915 - 2.469098805e-15 I,
        (7, 4, 4) = 11.64466094 - 7.004046619e-15 I,
        (7, 4, 5) = 5.0,
        (7, 4, 6) = 11.64466094 - 6.021812484e-15 I,
        (7, 4, 7) = 27.6862915 - 2.469098805e-15 I,
        (7, 4, 8) = 43.72792206 + 2.065849009e-15 I,
        (7, 5, 1) = 41.0,
        (7, 5, 2) = 35.72792206 + 1.776356839e-15 I,
        (7, 5, 3) = 23.0,
        (7, 5, 4) = 10.27207794 - 3.44386997e-15 I,
        (7, 5, 5) = 5.0,
        (7, 5, 6) = 10.27207794 - 1.776356839e-15 I,
        (7, 5, 7) = 23.0,
        (7, 5, 8) = 35.72792206 + 3.44386997e-15 I,
        (7, 6, 1) = 50.372583 - 3.552713679e-15 I,
        (7, 6, 2) = 43.72792206 + 1.776356839e-15 I,
        (7, 6, 3) = 27.6862915 - 1.776356839e-15 I,
        (7, 6, 4) = 11.64466094 - 6.311304654e-15 I,
        (7, 6, 5) = 5.0,
        (7, 6, 6) = 11.64466094 - 5.329070518e-15 I,
        (7, 6, 7) = 27.6862915 - 1.776356839e-15 I,
        (7, 6, 8) = 43.72792206 + 2.758590975e-15 I,
        (7, 7, 1) = 73.0,
        (7, 7, 2) = 63.04163056 + 3.552713679e-15 I,
        (7, 7, 3) = 39.0,
        (7, 7, 4) = 14.95836944 - 5.024790356e-15 I,
        (7, 7, 5) = 5.0,
        (7, 7, 6) = 14.95836944 - 3.552713679e-15 I,
        (7, 7, 7) = 39.0,
        (7, 7, 8) = 63.04163056 + 5.024790356e-15 I,
        (7, 8, 1) = 95.627417 + 4.93819761e-15 I,
        (7, 8, 2) = 82.35533906 + 9.574526163e-15 I,
        (7, 8, 3) = 50.3137085 + 2.469098805e-15 I,
        (7, 8, 4) = 18.27207794 - 6.598247771e-15 I,
        (7, 8, 5) = 5.0,
        (7, 8, 6) = 18.27207794 - 4.636328553e-15 I,
        (7, 8, 7) = 50.3137085 + 2.469098805e-15 I,
        (7, 8, 8) = 82.35533906 + 1.153644538e-14 I,
        (8, 1, 1) = 107.8284271 + 6.172747013e-16 I,
        (8, 1, 2) = 93.18376618 + 7.722702059e-15 I,
        (8, 1, 3) = 57.82842712 + 6.172747013e-16 I,
        (8, 1, 4) = 22.47308807 - 5.10025762e-15 I,
        (8, 1, 5) = 7.828427125 + 6.172747013e-16 I,
        (8, 1, 6) = 22.47308807 - 6.488152656e-15 I,
        (8, 1, 7) = 57.82842712 + 6.172747013e-16 I,
        (8, 1, 8) = 93.18376618 + 6.334807023e-15 I,
        (8, 2, 1) = 98.45584412 + 4.16998838e-15 I,
        (8, 2, 2) = 85.18376618 + 9.499058898e-15 I,
        (8, 2, 3) = 53.14213562 + 2.393631541e-15 I,
        (8, 2, 4) = 21.10050506 - 6.673715035e-15 I,
        (8, 2, 5) = 7.828427125 + 6.172747013e-16 I,
        (8, 2, 6) = 21.10050506 - 4.711795817e-15 I,
        (8, 2, 7) = 53.14213562 + 2.393631541e-15 I,
        (8, 2, 8) = 85.18376618 + 1.146097812e-14 I,
        (8, 3, 1) = 75.82842712 + 6.172747013e-16 I,
        (8, 3, 2) = 65.87005769 + 4.16998838e-15 I,
        (8, 3, 3) = 41.82842712 + 6.172747013e-16 I,
        (8, 3, 4) = 17.78679656 - 4.407515655e-15 I,
        (8, 3, 5) = 7.828427125 + 6.172747013e-16 I,
        (8, 3, 6) = 17.78679656 - 2.935438978e-15 I,
        (8, 3, 7) = 41.82842712 + 6.172747013e-16 I,
        (8, 3, 8) = 65.87005769 + 5.642065057e-15 I,
        (8, 4, 1) = 53.20101013 - 4.320922909e-15 I,
        (8, 4, 2) = 46.55634919 + 1.700889575e-15 I,
        (8, 4, 3) = 30.51471863 - 1.851824104e-15 I,
        (8, 4, 4) = 14.47308807 - 6.386771918e-15 I,
        (8, 4, 5) = 7.828427125 + 6.172747013e-16 I,
        (8, 4, 6) = 14.47308807 - 5.404537783e-15 I,
        (8, 4, 7) = 30.51471863 - 1.851824104e-15 I,
        (8, 4, 8) = 46.55634919 + 2.683123711e-15 I,
        (8, 5, 1) = 43.82842712 + 6.172747013e-16 I,
        (8, 5, 2) = 38.55634919 + 2.393631541e-15 I,
        (8, 5, 3) = 25.82842712 + 6.172747013e-16 I,
        (8, 5, 4) = 13.10050506 - 2.826595269e-15 I,
        (8, 5, 5) = 7.828427125 + 6.172747013e-16 I,
        (8, 5, 6) = 13.10050506 - 1.159082138e-15 I,
        (8, 5, 7) = 25.82842712 + 6.172747013e-16 I,
        (8, 5, 8) = 38.55634919 + 4.061144672e-15 I,
        (8, 6, 1) = 53.20101013 - 2.935438978e-15 I,
        (8, 6, 2) = 46.55634919 + 2.393631541e-15 I,
        (8, 6, 3) = 30.51471863 - 1.159082138e-15 I,
        (8, 6, 4) = 14.47308807 - 5.694029952e-15 I,
        (8, 6, 5) = 7.828427125 + 6.172747013e-16 I,
        (8, 6, 6) = 14.47308807 - 4.711795817e-15 I,
        (8, 6, 7) = 30.51471863 - 1.159082138e-15 I,
        (8, 6, 8) = 46.55634919 + 3.375865676e-15 I,
        (8, 7, 1) = 75.82842712 + 6.172747013e-16 I,
        (8, 7, 2) = 65.87005769 + 4.16998838e-15 I,
        (8, 7, 3) = 41.82842712 + 6.172747013e-16 I,
        (8, 7, 4) = 17.78679656 - 4.407515655e-15 I,
        (8, 7, 5) = 7.828427125 + 6.172747013e-16 I,
        (8, 7, 6) = 17.78679656 - 2.935438978e-15 I,
        (8, 7, 7) = 41.82842712 + 6.172747013e-16 I,
        (8, 7, 8) = 65.87005769 + 5.642065057e-15 I,
        (8, 8, 1) = 98.45584412 + 5.555472311e-15 I,
        (8, 8, 2) = 85.18376618 + 1.019180086e-14 I,
        (8, 8, 3) = 53.14213562 + 3.086373506e-15 I,
        (8, 8, 4) = 21.10050506 - 5.98097307e-15 I,
        (8, 8, 5) = 7.828427125 + 6.172747013e-16 I,
        (8, 8, 6) = 21.10050506 - 4.019053851e-15 I,
        (8, 8, 7) = 53.14213562 + 3.086373506e-15 I,
        (8, 8, 8) = 85.18376618 + 1.215372008e-14 I
      )