differential_equations/tableau/

verner.rs

//! Verner's Runge-Kutta methods with high-order interpolation. source: https://www.sfu.ca/~jverner/

use crate::{tableau::ButcherTableau, traits::Real};

impl<T: Real> ButcherTableau<T, 9, 10> {
    /// Verner's RKV(6,5,5)e method with 5th-order interpolation.
    ///
    ///
    /// A 'most efficient' Runge-Kutta (9,6(5)) FSAL pair with rational coefficients.
    ///
    /// # Overview
    /// This is a nine-stage FSAL pair of methods of orders p=6 and p=5, with dominant stage order 3.
    /// The coefficients are exact rationals. The method is "most efficient" in the sense that the
    /// maximum coefficient in `b` and `A` is not large, and the propagating formula nearly minimizes
    /// the 2-norm of the local truncation error (T_72 ≈ 1.44e-6).
    ///
    /// # Interpolation
    /// Additional stages and interpolating weights allow computation of an approximation at any point
    /// in the domain of solution of order up to p. These interpolants have continuous derivatives.
    ///
    /// The additional node c[9]=1/2 is chosen to achieve order 5 interpolation in ten stages, with
    /// maximum 2-norm of the local truncation error Ti_62 ≈ 3.18e-4 (three local maxima on [0,1]).
    ///
    /// The remaining two nodes are chosen to minimize the maximum 2-norm of the local truncation error
    /// for the order 6 interpolant, Ti_72 ≈ 4.44e-5 (two local maxima on [0,1]).
    ///
    /// # Source
    /// [Verner's RKV655e](https://www.sfu.ca/~jverner/RKV65.IIIXb.Efficient.00000144617.081204.RATOnWeb)
    ///
    pub fn rkv655e() -> Self {
        let mut c = [0.0; 10];
        let mut a = [[0.0; 10]; 10];
        let mut b = [0.0; 9];
        let mut bh = [0.0; 9];
        let mut bi5 = [[0.0; 10]; 10];

        c[0] = 0.0;
        c[1] = 0.6e-1;
        c[2] = 9.593_333_333_333_333e-2;
        c[3] = 0.1439;
        c[4] = 0.4973;
        c[5] = 0.9725;
        c[6] = 0.9995;
        c[7] = 1.0;
        c[8] = 1.0;

        a[1][0] = 0.6e-1;

        a[2][0] = 1.923_996_296_296_296_2e-2;
        a[2][1] = 7.669_337_037_037_037e-2;

        a[3][0] = 0.35975e-1;
        a[3][1] = 0.0;
        a[3][2] = 0.107925;

        a[4][0] = 1.318_683_415_233_148_4;
        a[4][1] = 0.0;
        a[4][2] = -5.042_058_063_628_562;
        a[4][3] = 4.220_674_648_395_414;

        a[5][0] = -41.872_591_664_327_516;
        a[5][1] = 0.0;
        a[5][2] = 159.432_562_163_137_5;
        a[5][3] = -122.119_213_565_010_03;
        a[5][4] = 5.531_743_066_200_054;

        a[6][0] = -54.430_156_935_316_504;
        a[6][1] = 0.0;
        a[6][2] = 207.067_251_365_018_48;
        a[6][3] = -158.610_813_784_59;
        a[6][4] = 6.991_816_585_950_242;
        a[6][5] = -1.859_723_106_220_323_4e-2;

        a[7][0] = -54.663_741_787_281_98;
        a[7][1] = 0.0;
        a[7][2] = 207.952_806_255_389_36;
        a[7][3] = -159.288_957_474_499_5;
        a[7][4] = 7.018_743_740_796_944;
        a[7][5] = -1.833_878_590_504_572_2e-2;
        a[7][6] = -5.119_484_997_882_099e-4;

        a[8][0] = 3.438_957_868_357_036e-2;
        a[8][1] = 0.0;
        a[8][2] = 0.0;
        a[8][3] = 0.258_262_455_563_350_3;
        a[8][4] = 0.420_937_118_967_353_7;
        a[8][5] = 4.405_396_469_669_31;
        a[8][6] = -176.483_119_024_298_65;
        a[8][7] = 172.364_133_401_415_07;

        b[0] = 3.438_957_868_357_036e-2;
        b[1] = 0.0;
        b[2] = 0.0;
        b[3] = 0.258_262_455_563_350_3;
        b[4] = 0.420_937_118_967_353_7;
        b[5] = 4.405_396_469_669_31;
        b[6] = -176.483_119_024_298_65;
        b[7] = 172.364_133_401_415_07;
        b[8] = 0.0;

        bh[0] = 4.909_967_648_382_49e-2;
        bh[1] = 0.0;
        bh[2] = 0.0;
        bh[3] = 0.225_111_222_951_652_42;
        bh[4] = 0.469_468_225_302_956_2;
        bh[5] = 0.806_579_224_998_886_8;
        bh[6] = 0.0;
        bh[7] = -0.607_119_489_177_796;
        bh[8] = 5.686_113_944_047_569_6e-2;

        c[9] = 0.5;

        a[9][0] = 0.016524159013572807;
        a[9][1] = 0.0;
        a[9][2] = 0.0;
        a[9][3] = 0.3053128187514179;
        a[9][4] = 0.207_120_093_820_197_9;
        a[9][5] = -1.293_879_140_655_123;
        a[9][6] = 57.11988411588149;
        a[9][7] = -55.87979207510932;
        a[9][8] = 0.024830028297766014;

        bi5[0][0] = 1.0;
        bi5[0][1] = -5.308169607103576;
        bi5[0][2] = 10.18168044895868;
        bi5[0][3] = -7.520036991611715;
        bi5[0][4] = 0.9340485368631161;
        bi5[0][5] = 0.746_867_191_577_065;

        bi5[1][0] = 0.0;
        bi5[1][1] = 0.0;
        bi5[1][2] = 0.0;
        bi5[1][3] = 0.0;
        bi5[1][4] = 0.0;
        bi5[1][5] = 0.0;

        bi5[2][0] = 0.0;
        bi5[2][1] = 0.0;
        bi5[2][2] = 0.0;
        bi5[2][3] = 0.0;
        bi5[2][4] = 0.0;
        bi5[2][5] = 0.0;

        bi5[3][0] = 0.0;
        bi5[3][1] = 6.272050253212501;
        bi5[3][2] = -16.02618147467746;
        bi5[3][3] = 12.844356324519617;
        bi5[3][4] = -1.148794504476759;
        bi5[3][5] = -1.6831681430145497;

        bi5[4][0] = 0.0;
        bi5[4][1] = 6.876491702846305;
        bi5[4][2] = -24.635767260846333;
        bi5[4][3] = 33.21078648379717;
        bi5[4][4] = -17.494_615_282_636_44;
        bi5[4][5] = 2.4640414758066497;

        bi5[5][0] = 0.0;
        bi5[5][1] = -35.5444517105996;
        bi5[5][2] = 165.7016170190242;
        bi5[5][3] = -385.463_539_549_114_3;
        bi5[5][4] = 442.4324137015702;
        bi5[5][5] = -182.7206429912112;

        bi5[6][0] = 0.0;
        bi5[6][1] = 1918.6548566980114;
        bi5[6][2] = -9268.121508966043;
        bi5[6][3] = 20858.33702877255;
        bi5[6][4] = -22645.82767158481;
        bi5[6][5] = 8960.474176055993;

        bi5[7][0] = 0.0;
        bi5[7][1] = -1883.0698021327183;
        bi5[7][2] = 9101.025187200634;
        bi5[7][3] = -20473.188551959537;
        bi5[7][4] = 22209.765551256534;
        bi5[7][5] = -8_782.168_250_963_5;

        bi5[8][0] = 0.0;
        bi5[8][1] = 0.11902479635123644;
        bi5[8][2] = -0.12502696705039375;
        bi5[8][3] = 1.779956919394999;
        bi5[8][4] = -4.660932123043763;
        bi5[8][5] = 2.886977374347921;

        bi5[9][0] = 0.0;
        bi5[9][1] = -8.0;
        bi5[9][2] = 32.0;
        bi5[9][3] = -40.0;
        bi5[9][4] = 16.0;
        bi5[9][5] = 0.0;

        let c = c.map(|x| T::from_f64(x).unwrap());
        let a = a.map(|row| row.map(|x| T::from_f64(x).unwrap()));
        let b = b.map(|x| T::from_f64(x).unwrap());
        let bh = bh.map(|x| T::from_f64(x).unwrap());
        let bi5 = bi5.map(|row| row.map(|x| T::from_f64(x).unwrap()));

        Self {
            c,
            a,
            b,
            bh: Some(bh),
            bi: Some(bi5),
            er: None,
        }
    }
}

impl<T: Real> ButcherTableau<T, 9, 12> {
    /// Verner's RKV(6,5,5)e method with 5th-order interpolation.
    ///
    ///
    /// A 'most efficient' Runge-Kutta (9,6(5)) FSAL pair with rational coefficients.
    ///
    /// # Overview
    /// This is a nine-stage FSAL pair of methods of orders p=6 and p=5, with dominant stage order 3.
    /// The coefficients are exact rationals. The method is "most efficient" in the sense that the
    /// maximum coefficient in `b` and `A` is not large, and the propagating formula nearly minimizes
    /// the 2-norm of the local truncation error (T_72 ≈ 1.44e-6).
    ///
    /// # Interpolation
    /// Additional stages and interpolating weights allow computation of an approximation at any point
    /// in the domain of solution of order up to p. These interpolants have continuous derivatives.
    ///
    /// The additional node c[9]=1/2 is chosen to achieve order 5 interpolation in ten stages, with
    /// maximum 2-norm of the local truncation error Ti_62 ≈ 3.18e-4 (three local maxima on [0,1]).
    ///
    /// The remaining two nodes are chosen to minimize the maximum 2-norm of the local truncation error
    /// for the order 6 interpolant, Ti_72 ≈ 4.44e-5 (two local maxima on [0,1]).
    ///
    /// # Source
    /// [Verner's RKV655e](https://www.sfu.ca/~jverner/RKV65.IIIXb.Efficient.00000144617.081204.RATOnWeb)
    ///
    pub fn rkv656e() -> Self {
        let mut c = [0.0; 12];
        let mut a = [[0.0; 12]; 12];
        let mut b = [0.0; 9];
        let mut bh = [0.0; 9];
        let mut bi6 = [[0.0; 12]; 12];

        c[0] = 0.0;
        c[1] = 0.6e-1;
        c[2] = 9.593_333_333_333_333e-2;
        c[3] = 0.1439;
        c[4] = 0.4973;
        c[5] = 0.9725;
        c[6] = 0.9995;
        c[7] = 1.0;
        c[8] = 1.0;

        a[1][0] = 0.6e-1;

        a[2][0] = 1.923_996_296_296_296_2e-2;
        a[2][1] = 7.669_337_037_037_037e-2;

        a[3][0] = 0.35975e-1;
        a[3][1] = 0.0;
        a[3][2] = 0.107925;

        a[4][0] = 1.318_683_415_233_148_4;
        a[4][1] = 0.0;
        a[4][2] = -5.042_058_063_628_562;
        a[4][3] = 4.220_674_648_395_414;

        a[5][0] = -41.872_591_664_327_516;
        a[5][1] = 0.0;
        a[5][2] = 159.432_562_163_137_5;
        a[5][3] = -122.119_213_565_010_03;
        a[5][4] = 5.531_743_066_200_054;

        a[6][0] = -54.430_156_935_316_504;
        a[6][1] = 0.0;
        a[6][2] = 207.067_251_365_018_48;
        a[6][3] = -158.610_813_784_59;
        a[6][4] = 6.991_816_585_950_242;
        a[6][5] = -1.859_723_106_220_323_4e-2;

        a[7][0] = -54.663_741_787_281_98;
        a[7][1] = 0.0;
        a[7][2] = 207.952_806_255_389_36;
        a[7][3] = -159.288_957_474_499_5;
        a[7][4] = 7.018_743_740_796_944;
        a[7][5] = -1.833_878_590_504_572_2e-2;
        a[7][6] = -5.119_484_997_882_099e-4;

        a[8][0] = 3.438_957_868_357_036e-2;
        a[8][1] = 0.0;
        a[8][2] = 0.0;
        a[8][3] = 0.258_262_455_563_350_3;
        a[8][4] = 0.420_937_118_967_353_7;
        a[8][5] = 4.405_396_469_669_31;
        a[8][6] = -176.483_119_024_298_65;
        a[8][7] = 172.364_133_401_415_07;

        b[0] = 3.438_957_868_357_036e-2;
        b[1] = 0.0;
        b[2] = 0.0;
        b[3] = 0.258_262_455_563_350_3;
        b[4] = 0.420_937_118_967_353_7;
        b[5] = 4.405_396_469_669_31;
        b[6] = -176.483_119_024_298_65;
        b[7] = 172.364_133_401_415_07;
        b[8] = 0.0;

        bh[0] = 4.909_967_648_382_49e-2;
        bh[1] = 0.0;
        bh[2] = 0.0;
        bh[3] = 0.225_111_222_951_652_42;
        bh[4] = 0.469_468_225_302_956_2;
        bh[5] = 0.806_579_224_998_886_8;
        bh[6] = 0.0;
        bh[7] = -0.607_119_489_177_796;
        bh[8] = 5.686_113_944_047_569_6e-2;

        c[9] = 0.5;

        a[9][0] = 0.016524159013572807;
        a[9][1] = 0.0;
        a[9][2] = 0.0;
        a[9][3] = 0.3053128187514179;
        a[9][4] = 0.207_120_093_820_197_9;
        a[9][5] = -1.293_879_140_655_123;
        a[9][6] = 57.11988411588149;
        a[9][7] = -55.87979207510932;
        a[9][8] = 0.024830028297766014;

        c[10] = 0.828;

        a[10][0] = 0.038150081818627746;
        a[10][1] = 0.0;
        a[10][2] = 0.0;
        a[10][3] = 0.250_235_825_251_370_5;
        a[10][4] = 0.3249441447817608;
        a[10][5] = 1.8224606658327962;
        a[10][6] = -67.7137233269262;
        a[10][7] = 66.03587911808127;
        a[10][8] = -0.0363881087495127;
        a[10][9] = 0.106441599909888;

        c[11] = 0.28;

        a[11][0] = 0.11178168039666012;
        a[11][1] = 0.0;
        a[11][2] = 0.0;
        a[11][3] = 0.025757505109345213;
        a[11][4] = 3.785140856363646;
        a[11][5] = 92.34088993695727;
        a[11][6] = -3819.461508432344;
        a[11][7] = 3732.492711530704;
        a[11][8] = -1.0756940209963033;
        a[11][9] = -3.231539970732086;
        a[11][10] = -4.707539085458635;

        bi6[0][0] = 1.0;
        bi6[0][1] = -7.778593856495575;
        bi6[0][2] = 27.0524385722671;
        bi6[0][3] = -45.78019011457698;
        bi6[0][4] = 36.72377741043638;
        bi6[0][5] = -11.183042432947356;

        bi6[1][0] = 0.0;
        bi6[1][1] = 0.0;
        bi6[1][2] = 0.0;
        bi6[1][3] = 0.0;
        bi6[1][4] = 0.0;
        bi6[1][5] = 0.0;

        bi6[2][0] = 0.0;
        bi6[2][1] = 0.0;
        bi6[2][2] = 0.0;
        bi6[2][3] = 0.0;
        bi6[2][4] = 0.0;
        bi6[2][5] = 0.0;

        bi6[3][0] = 0.0;
        bi6[3][1] = 16.63210213827976;
        bi6[3][2] = -86.25583404770622;
        bi6[3][3] = 171.7330546182696;
        bi6[3][4] = -149.6774409131595;
        bi6[3][5] = 47.82638065987969;

        bi6[4][0] = 0.0;
        bi6[4][1] = 27.10835046149758;
        bi6[4][2] = -140.58676162962996;
        bi6[4][3] = 279.904_475_796_891_7;
        bi6[4][4] = -243.95644583707966;
        bi6[4][5] = 77.95131832728772;

        bi6[5][0] = 0.0;
        bi6[5][1] = 283.70753264670357;
        bi6[5][2] = -1471.3371557366656;
        bi6[5][3] = 2929.3928569314393;
        bi6[5][4] = -2553.17199842168;
        bi6[5][5] = 815.8141610498723;

        bi6[6][0] = 0.0;
        bi6[6][1] = -11365.512865164834;
        bi6[6][2] = 58942.74718938947;
        bi6[6][3] = -117353.43045697975;
        bi6[6][4] = 102281.77209230464;
        bi6[6][5] = -32682.059078573825;

        bi6[7][0] = 0.0;
        bi6[7][1] = 11100.25019105113;
        bi6[7][2] = -57567.06701335558;
        bi6[7][3] = 114614.48808378985;
        bi6[7][4] = -99894.591091309;
        bi6[7][5] = 31919.283963225014;

        bi6[8][0] = 0.0;
        bi6[8][1] = -3.002282515073213;
        bi6[8][2] = 14.946122435958784;
        bi6[8][3] = -27.82695473251029;
        bi6[8][4] = 21.824672217437075;
        bi6[8][5] = -5.941557405812358;

        bi6[9][0] = 0.0;
        bi6[9][1] = -19.610347376201035;
        bi6[9][2] = 93.13370014508226;
        bi6[9][3] = -165.3493635542416;
        bi6[9][4] = 129.73901617804057;
        bi6[9][5] = -37.91300539268019;

        bi6[10][0] = 0.0;
        bi6[10][1] = -18.23029074639409;
        bi6[10][2] = 96.74593449012313;
        bi6[10][3] = -199.08634973839894;
        bi6[10][4] = 180.85605899200485;
        bi6[10][5] = -60.28535299733495;

        bi6[11][0] = 0.0;
        bi6[11][1] = -13.563796638614157;
        bi6[11][2] = 90.62137973668117;
        bi6[11][3] = -204.04515601697272;
        bi6[11][4] = 190.48135937835857;
        bi6[11][5] = -63.49378645945286;

        let c = c.map(|x| T::from_f64(x).unwrap());
        let a = a.map(|row| row.map(|x| T::from_f64(x).unwrap()));
        let b = b.map(|x| T::from_f64(x).unwrap());
        let bh = bh.map(|x| T::from_f64(x).unwrap());
        let bi6 = bi6.map(|row| row.map(|x| T::from_f64(x).unwrap()));

        Self {
            c,
            a,
            b,
            bh: Some(bh),
            bi: Some(bi6),
            er: None,
        }
    }
}

impl<T: Real> ButcherTableau<T, 10, 13> {
    /// A 'most efficient' Runge-Kutta (10:6(7)) pair with 6th-order interpolation.
    ///
    ///
    /// # Overview
    /// This provides a 10-stage Runge-Kutta method with:
    /// - Primary order: 6
    /// - Embedded order: 7 (for error estimation)
    /// - Number of stages: 10 primary + 3 additional stages for interpolation
    /// - Dominant stage order: 3
    ///
    /// # Efficiency
    /// This method is considered "most efficient" in the sense that for a specified
    /// maximum coefficient from `b` and `A`, it almost minimizes the local truncation
    /// error 2-norm:
    ///
    /// T_82 ~ 0.000003389
    ///
    /// (Formulas with slightly different nodes may have a slightly smaller error norm,
    /// perhaps achieved by having a larger maximum coefficient.)
    ///
    /// # Interpolation
    /// - Node c[11]=1 was chosen to make the interpolants differentiable
    /// - Node c[12] was chosen as a zero of a quadratic polynomial to enable efficient interpolation
    /// - The interpolant has continuous derivatives
    /// - Maximum 2-norm of the local truncation error over [0,1] for the interpolant:
    ///
    ///   Ti_72 ~ 0.0000165
    ///
    /// - The interpolation error has four local maximum values on [0,1]
    ///
    /// # Source
    /// [Verner's RKV766e](https://www.sfu.ca/~jverner/RKV76.IIa.Efficient.000003389335684.240711.FLOAT6040OnWeb)
    ///
    pub fn rkv766e() -> Self {
        let mut c = [0.0; 13];
        let mut a = [[0.0; 13]; 13];
        let mut b = [0.0; 10];
        let mut bh = [0.0; 10];
        let mut bi6 = [[0.0; 13]; 13];

        c[0] = 0.0;
        c[1] = 0.69e-1;
        c[2] = 0.118;
        c[3] = 0.177;
        c[4] = 0.501;
        c[5] = 0.773_779_911_530_533;
        c[6] = 0.994;
        c[7] = 0.998;
        c[8] = 1.0;
        c[9] = 1.0;

        a[1][0] = 0.69e-1;

        a[2][0] = 1.710_144_927_536_231_7e-2;
        a[2][1] = 0.100_898_550_724_637_69;

        a[3][0] = 0.4425e-1;
        a[3][1] = 0.0;
        a[3][2] = 0.13275;

        a[4][0] = 0.735_344_513_070_956_6;
        a[4][1] = 0.0;
        a[4][2] = -2.830_160_657_856_937_6;
        a[4][3] = 2.595_816_144_785_981;

        a[5][0] = -12.215_804_853_604_08;
        a[5][1] = 0.0;
        a[5][2] = 48.826_654_858_237_36;
        a[5][3] = -38.556_155_923_199_285;
        a[5][4] = 2.719_085_830_096_535_7;

        a[6][0] = 108.861_418_870_417_65;
        a[6][1] = 0.0;
        a[6][2] = -432.452_118_177_577_8;
        a[6][3] = 343.911_528_180_011_8;
        a[6][4] = -20.550_411_359_252_738;
        a[6][5] = 1.223_582_486_401_040_3;

        a[7][0] = 113.475_513_188_373_85;
        a[7][1] = 0.0;
        a[7][2] = -450.812_202_155_599_7;
        a[7][3] = 358.513_276_519_008_97;
        a[7][4] = -21.450_466_676_484_456;
        a[7][5] = 1.274_053_318_605_952_8;
        a[7][6] = -2.174_193_904_638_423e-3;

        a[8][0] = 115.699_622_332_423_26;
        a[8][1] = 0.0;
        a[8][2] = -459.663_544_610_024_8;
        a[8][3] = 365.553_471_713_174_6;
        a[8][4] = -21.885_115_863_497_848;
        a[8][5] = 1.298_718_109_698_721_5;
        a[8][6] = -5.318_700_918_481_884e-5;
        a[8][7] = -3.098_494_764_731_864_5e-3;

        a[9][0] = 124.154_393_561_246_47;
        a[9][1] = 0.0;
        a[9][2] = -493.231_871_331_459_73;
        a[9][3] = 392.208_621_931_580_07;
        a[9][4] = -23.486_415_642_908_533;
        a[9][5] = 1.362_322_948_908_907_5;
        a[9][6] = -7.051_467_367_205_771e-3;
        a[9][7] = 0.0;
        a[9][8] = 0.0;

        b[0] = 5.163_520_172_057_869e-2;
        b[1] = 0.0;
        b[2] = 0.0;
        b[3] = 0.276_717_253_546_164_9;
        b[4] = 0.337_417_528_528_715_1;
        b[5] = 0.188_448_826_781_096_78;
        b[6] = 24.541_341_216_348_68;
        b[7] = -68.811_902_844_690_12;
        b[8] = 44.416_342_817_764_885;
        b[9] = 0.0;

        bh[0] = 5.089_676_583_692_947_4e-2;
        bh[1] = 0.0;
        bh[2] = 0.0;
        bh[3] = 0.279_377_737_476_323_4;
        bh[4] = 0.328_133_014_274_653_5;
        bh[5] = 0.224_172_121_818_615_1;
        bh[6] = 0.787_457_477_801_507_7;
        bh[7] = 0.0;
        bh[8] = 0.0;
        bh[9] = -0.670_037_117_208_029_2;

        c[10] = 1.0;

        a[10][0] = 5.163_520_172_057_869e-2;
        a[10][1] = 0.0;
        a[10][2] = 0.0;
        a[10][3] = 0.276_717_253_546_164_9;
        a[10][4] = 0.337_417_528_528_715_1;
        a[10][5] = 0.188_448_826_781_096_78;
        a[10][6] = 24.541_341_216_348_68;
        a[10][7] = -68.811_902_844_690_12;
        a[10][8] = 44.416_342_817_764_885;
        a[10][9] = 0.0;

        c[11] = 0.320_691_974_286_432_15;

        a[11][0] = 5.595_947_882_055_416e-2;
        a[11][1] = 0.0;
        a[11][2] = 0.0;
        a[11][3] = 0.248_126_283_032_250_93;
        a[11][4] = 2.103_052_990_591_148e-2;
        a[11][5] = -1.086_652_415_454_008_5e-2;
        a[11][6] = 5.710_311_335_454_033;
        a[11][7] = -16.597_491_517_853_864;
        a[11][8] = 10.904_406_866_038_99;
        a[11][9] = 0.0;
        a[11][10] = -1.078_447_695_690_422_2e-2;

        c[12] = 0.107;

        a[12][0] = 5.365_176_199_729_764_6e-2;
        a[12][1] = 0.0;
        a[12][2] = 0.0;
        a[12][3] = 9.828_513_377_552_06e-2;
        a[12][4] = 2.236_190_536_978_546_8e-2;
        a[12][5] = -3.196_168_594_283_279_4e-3;
        a[12][6] = 0.545_100_325_730_724;
        a[12][7] = -1.638_732_002_009_432_7;
        a[12][8] = 1.093_896_392_979_467;
        a[12][9] = 0.0;
        a[12][10] = -4.657_483_105_197_486e-5;
        a[12][11] = -6.432_077_441_802_662e-2;

        bi6[0][0] = 1.0;
        bi6[0][1] = -7.582_446_684_249_579;
        bi6[0][2] = 21.653_396_979_374_616;
        bi6[0][3] = -26.503_131_701_115_624;
        bi6[0][4] = 13.685_670_411_429_184;
        bi6[0][5] = -2.201_853_803_718_02;

        bi6[1][0] = 0.0;
        bi6[1][1] = 0.0;
        bi6[1][2] = 0.0;
        bi6[1][3] = 0.0;
        bi6[1][4] = 0.0;
        bi6[1][5] = 0.0;

        bi6[2][0] = 0.0;
        bi6[2][1] = 0.0;
        bi6[2][2] = 0.0;
        bi6[2][3] = 0.0;
        bi6[2][4] = 0.0;
        bi6[2][5] = 0.0;

        bi6[3][0] = 0.0;
        bi6[3][1] = 5.889_034_278_272_536;
        bi6[3][2] = -57.969_601_549_357_755;
        bi6[3][3] = 174.106_666_886_173_12;
        bi6[3][4] = -196.200_362_716_086_12;
        bi6[3][5] = 74.450_980_354_544_38;

        bi6[4][0] = 0.0;
        bi6[4][1] = 3.196_839_349_796_654;
        bi6[4][2] = -31.323_975_727_005_283;
        bi6[4][3] = 93.052_981_029_703_84;
        bi6[4][4] = -102.896_887_106_406_17;
        bi6[4][5] = 38.308_459_982_439_665;

        bi6[5][0] = 0.0;
        bi6[5][1] = -8.787_886_596_190_12e-2;
        bi6[5][2] = 1.013_797_983_077_119_3;
        bi6[5][3] = -4.100_173_002_382_458_5;
        bi6[5][4] = 6.641_160_480_067_745;
        bi6[5][5] = -3.278_457_768_019_407_6;

        bi6[6][0] = 0.0;
        bi6[6][1] = -208.397_209_920_409_64;
        bi6[6][2] = 2_077.909_119_755_835;
        bi6[6][3] = -6_428.966_291_404_446;
        bi6[6][4] = 7_605.042_110_521_116;
        bi6[6][5] = -3_021.046_387_735_748;

        bi6[7][0] = 0.0;
        bi6[7][1] = 594.359_329_222_875_1;
        bi6[7][2] = -5_925.388_971_699_224;
        bi6[7][3] = 18_326.521_937_386_66;
        bi6[7][4] = -21_667.185_693_635_285;
        bi6[7][5] = 8_602.881_495_880_287;

        bi6[8][0] = 0.0;
        bi6[8][1] = -386.881_191_563_493_5;
        bi6[8][2] = 3_856.672_838_888_456_7;
        bi6[8][3] = -11_926.201_514_729_724;
        bi6[8][4] = 14_096.407_335_954_642;
        bi6[8][5] = -5_595.581_125_732_116;

        bi6[9][0] = 0.0;
        bi6[9][1] = 0.0;
        bi6[9][2] = 0.0;
        bi6[9][3] = 0.0;
        bi6[9][4] = 0.0;
        bi6[9][5] = 0.0;

        bi6[10][0] = 0.0;
        bi6[10][1] = 0.611_109_939_702_319_6;
        bi6[10][2] = -6.151_956_371_109_29;
        bi6[10][3] = 19.458_736_994_636_64;
        bi6[10][4] = -23.906_044_634_754_69;
        bi6[10][5] = 9.988_154_071_525_02;

        bi6[11][0] = 0.0;
        bi6[11][1] = -8.007_634_773_786_194;
        bi6[11][2] = 74.600_313_924_034_05;
        bi6[11][3] = -199.674_829_477_265_75;
        bi6[11][4] = 207.579_256_277_574_13;
        bi6[11][5] = -74.497_105_950_556_23;

        bi6[12][0] = 0.0;
        bi6[12][1] = 6.900_049_017_254_163_5;
        bi6[12][2] = -11.014_962_184_080_556;
        bi6[12][3] = -27.694_381_982_239_55;
        bi6[12][4] = 60.833_454_447_704_11;
        bi6[12][5] = -29.024_159_298_638_17;

        let c = c.map(|x| T::from_f64(x).unwrap());
        let a = a.map(|row| row.map(|x| T::from_f64(x).unwrap()));
        let b = b.map(|x| T::from_f64(x).unwrap());
        let bh = bh.map(|x| T::from_f64(x).unwrap());
        let bi6 = bi6.map(|row| row.map(|x| T::from_f64(x).unwrap()));

        Self {
            c,
            a,
            b,
            bh: Some(bh),
            bi: Some(bi6),
            er: None,
        }
    }
}

impl<T: Real> ButcherTableau<T, 10, 16> {
    /// A 'most efficient' Runge-Kutta (10:7(6)) pair with 7th-order interpolation.
    ///
    /// # Overview
    /// This provides a 10-stage Runge-Kutta method with:
    /// - Primary order: 7
    /// - Embedded order: 6 (for error estimation)
    /// - Number of stages: 10 primary + 6 additional stages for interpolation
    /// - Dominant stage order: 3
    ///
    /// # Efficiency
    /// This method is considered "most efficient" in the sense that for a specified
    /// maximum coefficient from `b` and `A`, it almost minimizes the local truncation
    /// error 2-norm:
    ///
    /// T_82 ~ 0.000003389
    ///
    /// # Interpolation
    /// - Additional stages allow computation of approximations at any point with order up to 7
    /// - The interpolant has continuous derivatives
    /// - Maximum 2-norm of the local truncation error for the 7th-order interpolant:
    ///
    ///   Ti_82 ~ 0.000003389
    ///
    /// - The 2-norm is bounded by this value on [0,1] and is nearly monotone increasing
    /// - The 7th-order interpolant requires 6 additional stages (16 total)
    ///
    /// # Notes
    /// - This method uses the same primary stages as the 6th-order method (rkv766e)
    /// - The higher-order interpolant requires more additional stages
    ///
    /// # Source
    /// [Verner's RKV766e](https://www.sfu.ca/~jverner/RKV76.IIa.Efficient.000003389335684.240711.FLOAT6040OnWeb)
    ///
    pub fn rkv767e() -> Self {
        let mut c = [0.0; 16];
        let mut a = [[0.0; 16]; 16];
        let mut b = [0.0; 10];
        let mut bh = [0.0; 10];
        let mut bi7 = [[0.0; 16]; 16];

        c[0] = 0.0;
        c[1] = 0.69e-1;
        c[2] = 0.118;
        c[3] = 0.177;
        c[4] = 0.501;
        c[5] = 0.773_779_911_530_533;
        c[6] = 0.994;
        c[7] = 0.998;
        c[8] = 1.0;
        c[9] = 1.0;

        a[1][0] = 0.69e-1;

        a[2][0] = 1.710_144_927_536_231_7e-2;
        a[2][1] = 0.100_898_550_724_637_69;

        a[3][0] = 0.4425e-1;
        a[3][1] = 0.0;
        a[3][2] = 0.13275;

        a[4][0] = 0.735_344_513_070_956_6;
        a[4][1] = 0.0;
        a[4][2] = -2.830_160_657_856_937_6;
        a[4][3] = 2.595_816_144_785_981;

        a[5][0] = -12.215_804_853_604_08;
        a[5][1] = 0.0;
        a[5][2] = 48.826_654_858_237_36;
        a[5][3] = -38.556_155_923_199_285;
        a[5][4] = 2.719_085_830_096_535_7;

        a[6][0] = 108.861_418_870_417_65;
        a[6][1] = 0.0;
        a[6][2] = -432.452_118_177_577_8;
        a[6][3] = 343.911_528_180_011_8;
        a[6][4] = -20.550_411_359_252_738;
        a[6][5] = 1.223_582_486_401_040_3;

        a[7][0] = 113.475_513_188_373_85;
        a[7][1] = 0.0;
        a[7][2] = -450.812_202_155_599_7;
        a[7][3] = 358.513_276_519_008_97;
        a[7][4] = -21.450_466_676_484_456;
        a[7][5] = 1.274_053_318_605_952_8;
        a[7][6] = -2.174_193_904_638_423e-3;

        a[8][0] = 115.699_622_332_423_26;
        a[8][1] = 0.0;
        a[8][2] = -459.663_544_610_024_8;
        a[8][3] = 365.553_471_713_174_6;
        a[8][4] = -21.885_115_863_497_848;
        a[8][5] = 1.298_718_109_698_721_5;
        a[8][6] = -5.318_700_918_481_884e-5;
        a[8][7] = -3.098_494_764_731_864_5e-3;

        a[9][0] = 124.154_393_561_246_47;
        a[9][1] = 0.0;
        a[9][2] = -493.231_871_331_459_73;
        a[9][3] = 392.208_621_931_580_07;
        a[9][4] = -23.486_415_642_908_533;
        a[9][5] = 1.362_322_948_908_907_5;
        a[9][6] = -7.051_467_367_205_771e-3;
        a[9][7] = 0.0;
        a[9][8] = 0.0;

        b[0] = 5.163_520_172_057_869e-2;
        b[1] = 0.0;
        b[2] = 0.0;
        b[3] = 0.276_717_253_546_164_9;
        b[4] = 0.337_417_528_528_715_1;
        b[5] = 0.188_448_826_781_096_78;
        b[6] = 24.541_341_216_348_68;
        b[7] = -68.811_902_844_690_12;
        b[8] = 44.416_342_817_764_885;
        b[9] = 0.0;

        bh[0] = 5.089_676_583_692_947_4e-2;
        bh[1] = 0.0;
        bh[2] = 0.0;
        bh[3] = 0.279_377_737_476_323_4;
        bh[4] = 0.328_133_014_274_653_5;
        bh[5] = 0.224_172_121_818_615_1;
        bh[6] = 0.787_457_477_801_507_7;
        bh[7] = 0.0;
        bh[8] = 0.0;
        bh[9] = -0.670_037_117_208_029_2;

        c[10] = 1.0;

        a[10][0] = 5.163_520_172_057_869e-2;
        a[10][1] = 0.0;
        a[10][2] = 0.0;
        a[10][3] = 0.276_717_253_546_164_9;
        a[10][4] = 0.337_417_528_528_715_1;
        a[10][5] = 0.188_448_826_781_096_78;
        a[10][6] = 24.541_341_216_348_68;
        a[10][7] = -68.811_902_844_690_12;
        a[10][8] = 44.416_342_817_764_885;
        a[10][9] = 0.0;

        c[11] = 0.320_691_974_286_432_15;

        a[11][0] = 5.595_947_882_055_416e-2;
        a[11][1] = 0.0;
        a[11][2] = 0.0;
        a[11][3] = 0.248_126_283_032_250_93;
        a[11][4] = 2.103_052_990_591_148e-2;
        a[11][5] = -1.086_652_415_454_008_5e-2;
        a[11][6] = 5.710_311_335_454_033;
        a[11][7] = -16.597_491_517_853_864;
        a[11][8] = 10.904_406_866_038_99;
        a[11][9] = 0.0;
        a[11][10] = -1.078_447_695_690_422_2e-2;

        c[12] = 0.107;

        a[12][0] = 5.365_176_199_729_764_6e-2;
        a[12][1] = 0.0;
        a[12][2] = 0.0;
        a[12][3] = 9.828_513_377_552_06e-2;
        a[12][4] = 2.236_190_536_978_546_8e-2;
        a[12][5] = -3.196_168_594_283_279_4e-3;
        a[12][6] = 0.545_100_325_730_724;
        a[12][7] = -1.638_732_002_009_432_7;
        a[12][8] = 1.093_896_392_979_467;
        a[12][9] = 0.0;
        a[12][10] = -4.657_483_105_197_486e-5;
        a[12][11] = -6.432_077_441_802_662e-2;

        c[13] = 0.3;

        a[13][0] = 1.919_717_775_847_849_8e-2;
        a[13][1] = 0.0;
        a[13][2] = 0.0;
        a[13][3] = -4.739_427_173_175_534e-2;
        a[13][4] = -2.641_522_514_821_109_7e-2;
        a[13][5] = 7.054_089_167_217_366e-8;
        a[13][6] = 1.551_079_892_101_621_3;
        a[13][7] = -4.428_095_538_025_377;
        a[13][8] = 2.883_718_325_674_217;
        a[13][9] = 0.0;
        a[13][10] = -4.297_481_934_497_428e-3;
        a[13][11] = 0.126_264_430_058_858_98;
        a[13][12] = 0.225_942_620_705_773_12;

        c[14] = 0.549;

        a[14][0] = 6.125_242_269_008_585e-2;
        a[14][1] = 0.0;
        a[14][2] = 0.0;
        a[14][3] = 0.252_531_040_183_435_36;
        a[14][4] = 0.150_705_487_741_318_06;
        a[14][5] = 1.024_270_639_019_054_2e-2;
        a[14][6] = -6.438_444_503_804_216;
        a[14][7] = 18.448_709_040_954_04;
        a[14][8] = -12.035_927_432_731_748;
        a[14][9] = 0.0;
        a[14][10] = 1.513_193_748_819_522_2e-2;
        a[14][11] = 0.104_344_770_529_991_54;
        a[14][12] = -1.954_546_944_129_134_7e-2;
        a[14][13] = 0.0;

        c[15] = 0.757;

        a[15][0] = 8.959_466_056_806_525e-2;
        a[15][1] = 0.0;
        a[15][2] = 0.0;
        a[15][3] = 0.638_921_494_656_219_4;
        a[15][4] = 0.430_912_029_682_266_2;
        a[15][5] = 7.695_592_670_277_011e-2;
        a[15][6] = -7.189_894_742_007_33;
        a[15][7] = 21.036_455_896_587_956;
        a[15][8] = -13.861_406_033_435_276;
        a[15][9] = 0.0;
        a[15][10] = 8.282_681_779_726_606e-3;
        a[15][11] = -0.214_845_674_306_927_8;
        a[15][12] = -0.257_976_240_227_466_6;
        a[15][13] = 0.0;
        a[15][14] = 0.0;

        bi7[0][0] = 1.0;
        bi7[0][1] = -8.503_890_289_208_65;
        bi7[0][2] = 31.182_343_390_857_11;
        bi7[0][3] = -56.830_479_769_185_395;
        bi7[0][4] = 52.797_398_314_710_22;
        bi7[0][5] = -22.951_833_027_205_392;
        bi7[0][6] = 3.358_096_581_752_686;

        bi7[1][0] = 0.0;
        bi7[1][1] = 0.0;
        bi7[1][2] = 0.0;
        bi7[1][3] = 0.0;
        bi7[1][4] = 0.0;
        bi7[1][5] = 0.0;
        bi7[1][6] = 0.0;

        bi7[2][0] = 0.0;
        bi7[2][1] = 0.0;
        bi7[2][2] = 0.0;
        bi7[2][3] = 0.0;
        bi7[2][4] = 0.0;
        bi7[2][5] = 0.0;
        bi7[2][6] = 0.0;

        bi7[3][0] = 0.0;
        bi7[3][1] = 6.267_709_807_177_576;
        bi7[3][2] = -68.591_451_234_481_41;
        bi7[3][3] = 270.100_637_895_418_06;
        bi7[3][4] = -486.238_809_416_271_64;
        bi7[3][5] = 407.139_981_823_150_04;
        bi7[3][6] = -128.401_351_621_446_42;

        bi7[4][0] = 0.0;
        bi7[4][1] = 7.642_585_077_624_11;
        bi7[4][2] = -83.637_567_434_426_29;
        bi7[4][3] = 329.349_502_153_457_14;
        bi7[4][4] = -592.899_413_554_681_6;
        bi7[4][5] = 496.449_587_698_277_47;
        bi7[4][6] = -156.567_276_411_722_08;

        bi7[5][0] = 0.0;
        bi7[5][1] = 4.268_409_521_381_499;
        bi7[5][2] = -46.711_863_270_912_34;
        bi7[5][3] = 183.942_806_861_248_7;
        bi7[5][4] = -331.136_320_542_610_63;
        bi7[5][5] = 277.268_767_765_684_74;
        bi7[5][6] = -87.443_351_508_010_85;

        bi7[6][0] = 0.0;
        bi7[6][1] = 555.867_055_818_904_2;
        bi7[6][2] = -6_083.199_322_405_53;
        bi7[6][3] = 23_954.530_599_007_394;
        bi7[6][4] = -43_123.268_902_078_34;
        bi7[6][5] = 36_108.197_406_176_53;
        bi7[6][6] = -11_387.585_495_302_614;

        bi7[7][0] = 0.0;
        bi7[7][1] = -1_558.605_518_026_587_3;
        bi7[7][2] = 17_056.790_705_448_41;
        bi7[7][3] = -67_166.534_124_506_99;
        bi7[7][4] = 120_914.100_165_743;
        bi7[7][5] = -101_244.416_509_538_55;
        bi7[7][6] = 31_929.853_378_036_034;

        bi7[8][0] = 0.0;
        bi7[8][1] = 1_006.040_439_872_400_5;
        bi7[8][2] = -11_009.726_980_722_82;
        bi7[8][3] = 43_354.298_925_413_49;
        bi7[8][4] = -78_046.993_360_795_04;
        bi7[8][5] = 65_350.710_068_603_25;
        bi7[8][6] = -20_609.912_749_553_514;

        bi7[9][0] = 0.0;
        bi7[9][1] = 0.0;
        bi7[9][2] = 0.0;
        bi7[9][3] = 0.0;
        bi7[9][4] = 0.0;
        bi7[9][5] = 0.0;
        bi7[9][6] = 0.0;

        bi7[10][0] = 0.0;
        bi7[10][1] = -1.612_970_387_190_759_7;
        bi7[10][2] = 17.729_097_727_353_25;
        bi7[10][3] = -70.349_373_843_121_95;
        bi7[10][4] = 128.135_008_891_280_17;
        bi7[10][5] = -109.073_435_226_653_7;
        bi7[10][6] = 35.171_672_838_332_98;

        bi7[11][0] = 0.0;
        bi7[11][1] = -2.626_599_644_929_272;
        bi7[11][2] = 9.920_024_380_612_826;
        bi7[11][3] = -7.644_130_389_085_102;
        bi7[11][4] = -14.864_305_527_142_132;
        bi7[11][5] = 26.113_902_923_734_624;
        bi7[11][6] = -10.898_891_743_190_944;

        bi7[12][0] = 0.0;
        bi7[12][1] = 9.561_675_242_924_823;
        bi7[12][2] = -36.112_108_563_035_484;
        bi7[12][3] = 27.827_115_729_688_02;
        bi7[12][4] = 54.110_896_739_260_276;
        bi7[12][5] = -95.063_082_630_066_82;
        bi7[12][6] = 39.675_503_481_229_17;

        bi7[13][0] = 0.0;
        bi7[13][1] = -9.791_432_054_831_755;
        bi7[13][2] = 116.673_354_759_619_02;
        bi7[13][3] = -427.017_866_152_982_6;
        bi7[13][4] = 710.069_381_780_811_7;
        bi7[13][5] = -556.821_423_866_992_9;
        bi7[13][6] = 166.887_985_534_376_5;

        bi7[14][0] = 0.0;
        bi7[14][1] = 7.372_370_767_192_203e-2;
        bi7[14][2] = 2.611_540_012_355_412;
        bi7[14][3] = -30.678_467_708_669_69;
        bi7[14][4] = 100.428_590_031_891_09;
        bi7[14][5] = -119.636_555_525_554_36;
        bi7[14][6] = 47.201_169_482_305_63;

        bi7[15][0] = 0.0;
        bi7[15][1] = -8.581_188_645_336_741;
        bi7[15][2] = 93.072_227_912_000_57;
        bi7[15][3] = -360.995_144_690_658_4;
        bi7[15][4] = 635.759_670_413_128_9;
        bi7[15][5] = -517.916_875_175_601_3;
        bi7[15][6] = 158.661_310_186_466_92;

        let c = c.map(|x| T::from_f64(x).unwrap());
        let a = a.map(|row| row.map(|x| T::from_f64(x).unwrap()));
        let b = b.map(|x| T::from_f64(x).unwrap());
        let bh = bh.map(|x| T::from_f64(x).unwrap());
        let bi7 = bi7.map(|row| row.map(|x| T::from_f64(x).unwrap()));

        Self {
            c,
            a,
            b,
            bh: Some(bh),
            bi: Some(bi7),
            er: None,
        }
    }
}

impl<T: Real> ButcherTableau<T, 13, 17> {
    /// An efficient Runge-Kutta (13:7) pair method with 7th-order interpolation.
    ///
    /// # Overview
    /// This provides a 13-stage Runge-Kutta method with:
    /// - Primary order: 7
    /// - Embedded order: 7 (used for stepsize control)
    /// - Number of stages: 13 primary + 4 additional stages for interpolation
    /// - Dominant stage order: 4
    ///
    /// # Interpolation
    /// - The method provides a 7th-order interpolant using the nodes c[14]=1, c[15]-c[17].
    /// - The interpolant has continuous derivatives.
    /// - Maximum 2-norm of the local truncation error over [0,1] for the interpolant:
    ///   
    ///   Ti_82 ~ 0.0000063833
    ///   
    /// - The interpolation error has three local maximum values on [0,1].
    ///
    /// # Notes
    /// - This method uses the same initial 13 stages as the 8th-order method (rkv878e).
    /// - The interpolant requires fewer additional stages than the 8th-order version.
    ///
    /// # Source
    /// [Verner's RKV878e](https://www.sfu.ca/~jverner/RKV87.IIa.Efficient.000000011182-240510.FLOAT6040OnWeb)
    pub fn rkv877e() -> Self {
        let mut c = [0.0; 17];
        let mut a = [[0.0; 17]; 17];
        let mut b = [0.0; 13];
        let mut bh = [0.0; 13];
        let mut bi7 = [[0.0; 17]; 17];

        c[0] = 0.0;
        c[1] = 0.92662e-1;
        c[2] = 0.131_223_036_175_401_76;
        c[3] = 0.196_834_554_263_102_65;
        c[4] = 0.427173;
        c[5] = 0.485972;
        c[6] = 0.161915;
        c[7] = 0.985468;
        c[8] = 0.962_697_734_860_454_1;
        c[9] = 0.99626;
        c[10] = 0.997947;
        c[11] = 1.0;
        c[12] = 1.0;

        a[1][0] = 0.92662e-1;

        a[2][0] = 3.830_746_548_250_284e-2;
        a[2][1] = 9.291_557_069_289_892e-2;

        a[3][0] = 4.920_863_856_577_566e-2;
        a[3][1] = 0.0;
        a[3][2] = 0.147_625_915_697_326_97;

        a[4][0] = 0.274_307_608_570_248_66;
        a[4][1] = 0.0;
        a[4][2] = -0.931_988_720_310_265_6;
        a[4][3] = 1.084_854_111_740_017;

        a[5][0] = 6.461_852_970_939_692e-2;
        a[5][1] = 0.0;
        a[5][2] = 0.0;
        a[5][3] = 0.268_762_921_336_892_34;
        a[5][4] = 0.152_590_548_953_710_74;

        a[6][0] = 7.189_155_819_773_217e-2;
        a[6][1] = 0.0;
        a[6][2] = 0.0;
        a[6][3] = 0.122_126_578_336_254_97;
        a[6][4] = -7.943_550_859_198_562e-2;
        a[6][5] = 4.733_237_205_799_848e-2;

        a[7][0] = -6.073_603_893_714_329;
        a[7][1] = 0.0;
        a[7][2] = 0.0;
        a[7][3] = -73.8956;
        a[7][4] = 11.939_853_706_952_74;
        a[7][5] = -3.839_251_541_405_054_5;
        a[7][6] = 72.854_069_728_166_65;

        a[8][0] = -4.868_640_079_323_569;
        a[8][1] = 0.0;
        a[8][2] = 0.0;
        a[8][3] = -59.185_727_999_756_46;
        a[8][4] = 9.230_819_319_232_426;
        a[8][5] = -2.676_847_914_962_525_7;
        a[8][6] = 58.457_200_099_946_86;
        a[8][7] = 5.894_309_723_726_360_4e-3;

        a[9][0] = -6.689_861_899_320_853;
        a[9][1] = 0.0;
        a[9][2] = 0.0;
        a[9][3] = -81.442_710_040_531_11;
        a[9][4] = 13.367_788_256_983_971;
        a[9][5] = -4.470_777_638_416_181;
        a[9][6] = 80.233_213_921_614_11;
        a[9][7] = -1.313_638_336_212_181_5e-2;
        a[9][8] = 1.174_378_303_219_413_9e-2;

        a[10][0] = -6.788_841_955_800_464;
        a[10][1] = 0.0;
        a[10][2] = 0.0;
        a[10][3] = -82.656_398_559_348_29;
        a[10][4] = 13.599_739_218_748_99;
        a[10][5] = -4.574_464_055_350_504;
        a[10][6] = 81.419_432_072_160_76;
        a[10][7] = -1.416_248_014_826_418e-2;
        a[10][8] = 1.375_441_580_835_227_5e-2;
        a[10][9] = -1.111_656_070_581_006_2e-3;

        a[11][0] = -6.910_189_846_402_486;
        a[11][1] = 0.0;
        a[11][2] = 0.0;
        a[11][3] = -84.144_951_541_767_48;
        a[11][4] = 13.885_121_223_789_838;
        a[11][5] = -4.702_458_788_144_494;
        a[11][6] = 82.874_114_515_292_42;
        a[11][7] = -1.645_498_337_198_78e-2;
        a[11][8] = 1.644_663_972_162_521_3e-2;
        a[11][9] = 4.275_449_370_796_531e-3;
        a[11][10] = -5.902_668_488_222_362e-3;

        a[12][0] = -6.911_973_921_198_98;
        a[12][1] = 0.0;
        a[12][2] = 0.0;
        a[12][3] = -84.166_355_958_787_8;
        a[12][4] = 13.888_346_275_655_82;
        a[12][5] = -4.703_463_178_409_702;
        a[12][6] = 82.895_186_222_074_05;
        a[12][7] = -1.020_345_016_228_260_3e-2;
        a[12][8] = 1.427_900_423_230_391_5e-2;
        a[12][9] = -5.814_993_403_397_982e-3;
        a[12][10] = 0.0;
        a[12][11] = 0.0;

        b[0] = 4.625_543_159_712_467_6e-2;
        b[1] = 0.0;
        b[2] = 0.0;
        b[3] = 0.0;
        b[4] = 0.0;
        b[5] = 0.370_666_616_552_101_1;
        b[6] = 0.259_044_082_455_274_66;
        b[7] = -679.984_146_817_503_9;
        b[8] = 49.891_611_290_420_535;
        b[9] = 10_271.235_222_137_313;
        b[10] = -14_782.196_606_356_898;
        b[11] = 5_141.377_953_616_064;
        b[12] = 0.0;

        bh[0] = 4.638_504_234_365_211e-2;
        bh[1] = 0.0;
        bh[2] = 0.0;
        bh[3] = 0.0;
        bh[4] = 0.0;
        bh[5] = 0.372_576_768_158_119_6;
        bh[6] = 0.258_568_549_512_168_73;
        bh[7] = -147.495_076_758_926_53;
        bh[8] = 23.843_627_126_445_874;
        bh[9] = 347.426_416_673_055_1;
        bh[10] = 0.0;
        bh[11] = 0.0;
        bh[12] = -223.452_497_400_588_36;

        c[13] = 1.0;

        a[13][0] = 4.625_543_159_712_467_6e-2;
        a[13][1] = 0.0;
        a[13][2] = 0.0;
        a[13][3] = 0.0;
        a[13][4] = 0.0;
        a[13][5] = 0.370_666_616_552_101_1;
        a[13][6] = 0.259_044_082_455_274_66;
        a[13][7] = -679.984_146_817_503_9;
        a[13][8] = 49.891_611_290_420_535;
        a[13][9] = 10_271.235_222_137_313;
        a[13][10] = -14_782.196_606_356_898;
        a[13][11] = 5_141.377_953_616_064;
        a[13][12] = 0.0;

        c[14] = 0.311_017_763_495_386_4;

        a[14][0] = 4.911_138_760_562_453e-2;
        a[14][1] = 0.0;
        a[14][2] = 0.0;
        a[14][3] = 0.0;
        a[14][4] = 0.0;
        a[14][5] = 3.117_373_122_649_833e-2;
        a[14][6] = 0.238_492_367_955_329;
        a[14][7] = -114.318_025_626_252_7;
        a[14][8] = 3.546_408_122_969_840_5;
        a[14][9] = 2_460.597_003_929_216;
        a[14][10] = -3_738.924_986_857_494;
        a[14][11] = 1_389.104_435_743_812_5;
        a[14][12] = 0.0;
        a[14][13] = -1.259_503_554_386_166_3e-2;

        c[15] = 0.2;

        a[15][0] = 5.179_349_197_485_644e-2;
        a[15][1] = 0.0;
        a[15][2] = 0.0;
        a[15][3] = 0.0;
        a[15][4] = 0.0;
        a[15][5] = 3.491_876_693_023_422e-2;
        a[15][6] = 0.199_195_855_918_545_47;
        a[15][7] = -83.397_932_805_325_08;
        a[15][8] = 3.594_107_171_680_368_3;
        a[15][9] = 1_608.071_558_743_333_5;
        a[15][10] = -2_405.179_666_082_838_6;
        a[15][11] = 876.913_423_768_755_8;
        a[15][12] = 0.0;
        a[15][13] = -7.325_540_972_670_705e-3;
        a[15][14] = -8.007_336_945_700_194e-2;

        c[16] = 0.6;

        a[16][0] = 4.436_713_671_328_35e-2;
        a[16][1] = 0.0;
        a[16][2] = 0.0;
        a[16][3] = 0.0;
        a[16][4] = 0.0;
        a[16][5] = 0.234_106_144_170_543_85;
        a[16][6] = 0.361_391_276_009_148_67;
        a[16][7] = 148.988_450_648_249_68;
        a[16][8] = -9.426_031_101_789_148;
        a[16][9] = -2_394.447_533_524_453_3;
        a[16][10] = 3_478.192_564_092_670_6;
        a[16][11] = -1_223.319_642_639_490_7;
        a[16][12] = 0.0;
        a[16][13] = 1.180_395_533_992_265_7e-2;
        a[16][14] = 0.134_964_703_701_482_84;
        a[16][15] = -0.174_440_691_121_442_12;

        bi7[0][0] = 1.0;
        bi7[0][1] = -8.212_110_712_910_063;
        bi7[0][2] = 35.026_353_207_696_05;
        bi7[0][3] = -83.002_133_398_763_38;
        bi7[0][4] = 108.640_900_005_722_67;
        bi7[0][5] = -72.996_471_060_209_22;
        bi7[0][6] = 19.589_717_390_061_065;

        bi7[1][0] = 0.0;
        bi7[1][1] = 0.0;
        bi7[1][2] = 0.0;
        bi7[1][3] = 0.0;
        bi7[1][4] = 0.0;
        bi7[1][5] = 0.0;
        bi7[1][6] = 0.0;

        bi7[2][0] = 0.0;
        bi7[2][1] = 0.0;
        bi7[2][2] = 0.0;
        bi7[2][3] = 0.0;
        bi7[2][4] = 0.0;
        bi7[2][5] = 0.0;
        bi7[2][6] = 0.0;

        bi7[3][0] = 0.0;
        bi7[3][1] = 0.0;
        bi7[3][2] = 0.0;
        bi7[3][3] = 0.0;
        bi7[3][4] = 0.0;
        bi7[3][5] = 0.0;
        bi7[3][6] = 0.0;

        bi7[4][0] = 0.0;
        bi7[4][1] = 0.0;
        bi7[4][2] = 0.0;
        bi7[4][3] = 0.0;
        bi7[4][4] = 0.0;
        bi7[4][5] = 0.0;
        bi7[4][6] = 0.0;

        bi7[5][0] = 0.0;
        bi7[5][1] = 24.495_046_326_791_247;
        bi7[5][2] = -197.698_434_568_493_92;
        bi7[5][3] = 648.816_707_921_001_5;
        bi7[5][4] = -1_029.913_614_541_902;
        bi7[5][5] = 784.290_278_276_683_6;
        bi7[5][6] = -229.619_316_797_528_34;

        bi7[6][0] = 0.0;
        bi7[6][1] = 31.569_189_535_057_046;
        bi7[6][2] = -253.501_865_152_415_8;
        bi7[6][3] = 825.652_427_431_980_9;
        bi7[6][4] = -1_296.622_633_293_770_4;
        bi7[6][5] = 974.262_805_803_163_1;
        bi7[6][6] = -281.100_880_241_559_54;

        bi7[7][0] = 0.0;
        bi7[7][1] = 13_532.460_624_863_495;
        bi7[7][2] = -103_993.691_210_812_35;
        bi7[7][3] = 315_790.051_792_644_2;
        bi7[7][4] = -444_650.940_863_563_4;
        bi7[7][5] = 285_484.299_040_403_7;
        bi7[7][6] = -66_842.163_530_353_08;

        bi7[8][0] = 0.0;
        bi7[8][1] = -797.337_393_774_812_7;
        bi7[8][2] = 6_069.299_906_337_187;
        bi7[8][3] = -18_132.728_486_459_506;
        bi7[8][4] = 24_818.083_374_433_99;
        bi7[8][5] = -15_179.252_666_931_208;
        bi7[8][6] = 3_271.826_877_684_767_6;

        bi7[9][0] = 0.0;
        bi7[9][1] = -223_490.905_898_175_5;
        bi7[9][2] = 1_723_137.281_290_412;
        bi7[9][3] = -5_261_550.360_871_224;
        bi7[9][4] = 7_478_225.628_671_212;
        bi7[9][5] = -4_874_996.123_844_563;
        bi7[9][6] = 1_168_945.715_874_475_4;

        bi7[10][0] = 0.0;
        bi7[10][1] = 325_937.354_598_091_16;
        bi7[10][2] = -2_514_175.045_366_639;
        bi7[10][3] = 7_682_914.240_892_724;
        bi7[10][4] = -10_933_909.132_147_21;
        bi7[10][5] = 7_142_613.573_847_849;
        bi7[10][6] = -1_718_163.188_431_172_2;

        bi7[11][0] = 0.0;
        bi7[11][1] = -115_180.893_543_081_68;
        bi7[11][2] = 888_954.826_743_313_3;
        bi7[11][3] = -2_718_988.108_015_552_6;
        bi7[11][4] = 3_875_443.744_975_928;
        bi7[11][5] = -2_537_848.359_487_731;
        bi7[11][6] = 612_760.167_280_739_9;

        bi7[12][0] = 0.0;
        bi7[12][1] = 0.0;
        bi7[12][2] = 0.0;
        bi7[12][3] = 0.0;
        bi7[12][4] = 0.0;
        bi7[12][5] = 0.0;
        bi7[12][6] = 0.0;

        bi7[13][0] = 0.0;
        bi7[13][1] = 0.817_197_452_808_954;
        bi7[13][2] = -5.929_889_493_933_239;
        bi7[13][3] = 16.170_471_422_259_21;
        bi7[13][4] = -18.188_640_730_913_09;
        bi7[13][5] = 6.499_437_906_736_728;
        bi7[13][6] = 0.631_423_443_041_433_8;

        bi7[14][0] = 0.0;
        bi7[14][1] = -19.589_917_077_606_923;
        bi7[14][2] = 128.442_052_186_315_6;
        bi7[14][3] = -321.875_628_679_264_2;
        bi7[14][4] = 394.572_135_120_649_14;
        bi7[14][5] = -239.336_007_560_733_55;
        bi7[14][6] = 57.787_366_010_639_886;

        bi7[15][0] = 0.0;
        bi7[15][1] = -15.352_746_937_526_659;
        bi7[15][2] = 180.168_624_955_189_53;
        bi7[15][3] = -701.647_641_421_898_2;
        bi7[15][4] = 1_211.468_823_272_660_8;
        bi7[15][5] = -961.905_487_412_752;
        bi7[15][6] = 287.268_427_544_326_4;

        bi7[16][0] = 0.0;
        bi7[16][1] = -14.405_046_509_284_272;
        bi7[16][2] = 120.821_796_254_251_92;
        bi7[16][3] = -417.209_515_408_200_8;
        bi7[16][4] = 702.659_019_366_941;
        bi7[16][5] = -564.951_444_979_866;
        bi7[16][6] = 173.085_191_276_158_13;

        let c = c.map(|x| T::from_f64(x).unwrap());
        let a = a.map(|row| row.map(|x| T::from_f64(x).unwrap()));
        let b = b.map(|x| T::from_f64(x).unwrap());
        let bh = bh.map(|x| T::from_f64(x).unwrap());
        let bi7 = bi7.map(|row| row.map(|x| T::from_f64(x).unwrap()));

        Self {
            c,
            a,
            b,
            bh: Some(bh),
            bi: Some(bi7),
            er: None,
        }
    }
}

impl<T: Real> ButcherTableau<T, 13, 21> {
    /// A highly efficient Runge-Kutta (13:8(7)) pair method.
    ///
    /// # Overview
    /// This is an efficient 13-stage Runge-Kutta method that provides:
    /// - Primary order: 8
    /// - Embedded order: 7 (for error estimation)
    /// - Number of stages: 13 primary + 8 additional stages for interpolation
    /// - Dominant stage order: 4
    ///
    /// # Efficiency
    /// The method is particularly "efficient" with a local truncation error 2-norm:
    ///
    /// T_92 ~ 0.000000011182
    ///
    /// which appears to be the minimum possible for this type of method.
    ///
    /// # Interpolation
    /// - The additional stages (c[14] through c[21]) allow computation of approximations
    ///   at any point in the solution domain with order 8.
    /// - The interpolant has continuous derivatives.
    /// - Maximum 2-norm of the local truncation error over [0,1] for the interpolant:
    ///   
    ///   Ti_92 ~ 0.0000011515
    ///   
    /// - The interpolation error has two local maximum values on [0,1].
    ///
    /// # Source
    /// [Verner's RKV878e](https://www.sfu.ca/~jverner/RKV87.IIa.Efficient.000000011182-240510.FLOAT6040OnWeb)
    pub fn rkv878e() -> Self {
        let mut c = [0.0; 21];
        let mut a = [[0.0; 21]; 21];
        let mut b = [0.0; 13];
        let mut bh = [0.0; 13];
        let mut bi8 = [[0.0; 21]; 21];

        c[0] = 0.0;
        c[1] = 0.92662e-1;
        c[2] = 0.131_223_036_175_401_76;
        c[3] = 0.196_834_554_263_102_65;
        c[4] = 0.427173;
        c[5] = 0.485972;
        c[6] = 0.161915;
        c[7] = 0.985468;
        c[8] = 0.962_697_734_860_454_1;
        c[9] = 0.99626;
        c[10] = 0.997947;
        c[11] = 1.0;
        c[12] = 1.0;

        a[1][0] = 0.92662e-1;

        a[2][0] = 3.830_746_548_250_284e-2;
        a[2][1] = 9.291_557_069_289_892e-2;

        a[3][0] = 4.920_863_856_577_566e-2;
        a[3][1] = 0.0;
        a[3][2] = 0.147_625_915_697_326_97;

        a[4][0] = 0.274_307_608_570_248_66;
        a[4][1] = 0.0;
        a[4][2] = -0.931_988_720_310_265_6;
        a[4][3] = 1.084_854_111_740_017;

        a[5][0] = 6.461_852_970_939_692e-2;
        a[5][1] = 0.0;
        a[5][2] = 0.0;
        a[5][3] = 0.268_762_921_336_892_34;
        a[5][4] = 0.152_590_548_953_710_74;

        a[6][0] = 7.189_155_819_773_217e-2;
        a[6][1] = 0.0;
        a[6][2] = 0.0;
        a[6][3] = 0.122_126_578_336_254_97;
        a[6][4] = -7.943_550_859_198_562e-2;
        a[6][5] = 4.733_237_205_799_848e-2;

        a[7][0] = -6.073_603_893_714_329;
        a[7][1] = 0.0;
        a[7][2] = 0.0;
        a[7][3] = -73.8956;
        a[7][4] = 11.939_853_706_952_74;
        a[7][5] = -3.839_251_541_405_054_5;
        a[7][6] = 72.854_069_728_166_65;

        a[8][0] = -4.868_640_079_323_569;
        a[8][1] = 0.0;
        a[8][2] = 0.0;
        a[8][3] = -59.185_727_999_756_46;
        a[8][4] = 9.230_819_319_232_426;
        a[8][5] = -2.676_847_914_962_525_7;
        a[8][6] = 58.457_200_099_946_86;
        a[8][7] = 5.894_309_723_726_360_4e-3;

        a[9][0] = -6.689_861_899_320_853;
        a[9][1] = 0.0;
        a[9][2] = 0.0;
        a[9][3] = -81.442_710_040_531_11;
        a[9][4] = 13.367_788_256_983_971;
        a[9][5] = -4.470_777_638_416_181;
        a[9][6] = 80.233_213_921_614_11;
        a[9][7] = -1.313_638_336_212_181_5e-2;
        a[9][8] = 1.174_378_303_219_413_9e-2;

        a[10][0] = -6.788_841_955_800_464;
        a[10][1] = 0.0;
        a[10][2] = 0.0;
        a[10][3] = -82.656_398_559_348_29;
        a[10][4] = 13.599_739_218_748_99;
        a[10][5] = -4.574_464_055_350_504;
        a[10][6] = 81.419_432_072_160_76;
        a[10][7] = -1.416_248_014_826_418e-2;
        a[10][8] = 1.375_441_580_835_227_5e-2;
        a[10][9] = -1.111_656_070_581_006_2e-3;

        a[11][0] = -6.910_189_846_402_486;
        a[11][1] = 0.0;
        a[11][2] = 0.0;
        a[11][3] = -84.144_951_541_767_48;
        a[11][4] = 13.885_121_223_789_838;
        a[11][5] = -4.702_458_788_144_494;
        a[11][6] = 82.874_114_515_292_42;
        a[11][7] = -1.645_498_337_198_78e-2;
        a[11][8] = 1.644_663_972_162_521_3e-2;
        a[11][9] = 4.275_449_370_796_531e-3;
        a[11][10] = -5.902_668_488_222_362e-3;

        a[12][0] = -6.911_973_921_198_98;
        a[12][1] = 0.0;
        a[12][2] = 0.0;
        a[12][3] = -84.166_355_958_787_8;
        a[12][4] = 13.888_346_275_655_82;
        a[12][5] = -4.703_463_178_409_702;
        a[12][6] = 82.895_186_222_074_05;
        a[12][7] = -1.020_345_016_228_260_3e-2;
        a[12][8] = 1.427_900_423_230_391_5e-2;
        a[12][9] = -5.814_993_403_397_982e-3;
        a[12][10] = 0.0;
        a[12][11] = 0.0;

        b[0] = 4.625_543_159_712_467_6e-2;
        b[1] = 0.0;
        b[2] = 0.0;
        b[3] = 0.0;
        b[4] = 0.0;
        b[5] = 0.370_666_616_552_101_1;
        b[6] = 0.259_044_082_455_274_66;
        b[7] = -679.984_146_817_503_9;
        b[8] = 49.891_611_290_420_535;
        b[9] = 10_271.235_222_137_313;
        b[10] = -14_782.196_606_356_898;
        b[11] = 5_141.377_953_616_064;
        b[12] = 0.0;

        bh[0] = 4.638_504_234_365_211e-2;
        bh[1] = 0.0;
        bh[2] = 0.0;
        bh[3] = 0.0;
        bh[4] = 0.0;
        bh[5] = 0.372_576_768_158_119_6;
        bh[6] = 0.258_568_549_512_168_73;
        bh[7] = -147.495_076_758_926_53;
        bh[8] = 23.843_627_126_445_874;
        bh[9] = 347.426_416_673_055_1;
        bh[10] = 0.0;
        bh[11] = 0.0;
        bh[12] = -223.452_497_400_588_36;

        c[13] = 1.0;

        a[13][0] = 4.625_543_159_712_467_6e-2;
        a[13][1] = 0.0;
        a[13][2] = 0.0;
        a[13][3] = 0.0;
        a[13][4] = 0.0;
        a[13][5] = 0.370_666_616_552_101_1;
        a[13][6] = 0.259_044_082_455_274_66;
        a[13][7] = -679.984_146_817_503_9;
        a[13][8] = 49.891_611_290_420_535;
        a[13][9] = 10_271.235_222_137_313;
        a[13][10] = -14_782.196_606_356_898;
        a[13][11] = 5_141.377_953_616_064;
        a[13][12] = 0.0;

        c[14] = 0.311_017_763_495_386_4;

        a[14][0] = 4.911_138_760_562_453e-2;
        a[14][1] = 0.0;
        a[14][2] = 0.0;
        a[14][3] = 0.0;
        a[14][4] = 0.0;
        a[14][5] = 3.117_373_122_649_833e-2;
        a[14][6] = 0.238_492_367_955_329;
        a[14][7] = -114.318_025_626_252_7;
        a[14][8] = 3.546_408_122_969_840_5;
        a[14][9] = 2_460.597_003_929_216;
        a[14][10] = -3_738.924_986_857_494;
        a[14][11] = 1_389.104_435_743_812_5;
        a[14][12] = 0.0;
        a[14][13] = -1.259_503_554_386_166_3e-2;

        c[15] = 0.2;

        a[15][0] = 5.179_349_197_485_644e-2;
        a[15][1] = 0.0;
        a[15][2] = 0.0;
        a[15][3] = 0.0;
        a[15][4] = 0.0;
        a[15][5] = 3.491_876_693_023_422e-2;
        a[15][6] = 0.199_195_855_918_545_47;
        a[15][7] = -83.397_932_805_325_08;
        a[15][8] = 3.594_107_171_680_368_3;
        a[15][9] = 1_608.071_558_743_333_5;
        a[15][10] = -2_405.179_666_082_838_6;
        a[15][11] = 876.913_423_768_755_8;
        a[15][12] = 0.0;
        a[15][13] = -7.325_540_972_670_705e-3;
        a[15][14] = -8.007_336_945_700_194e-2;

        c[16] = 0.6;

        a[16][0] = 4.436_713_671_328_35e-2;
        a[16][1] = 0.0;
        a[16][2] = 0.0;
        a[16][3] = 0.0;
        a[16][4] = 0.0;
        a[16][5] = 0.234_106_144_170_543_85;
        a[16][6] = 0.361_391_276_009_148_67;
        a[16][7] = 148.988_450_648_249_68;
        a[16][8] = -9.426_031_101_789_148;
        a[16][9] = -2_394.447_533_524_453_3;
        a[16][10] = 3_478.192_564_092_670_6;
        a[16][11] = -1_223.319_642_639_490_7;
        a[16][12] = 0.0;
        a[16][13] = 1.180_395_533_992_265_7e-2;
        a[16][14] = 0.134_964_703_701_482_84;
        a[16][15] = -0.174_440_691_121_442_12;

        c[17] = 0.251;

        a[17][0] = 4.927_309_335_160_545e-2;
        a[17][1] = 0.0;
        a[17][2] = 0.0;
        a[17][3] = 0.0;
        a[17][4] = 0.0;
        a[17][5] = 0.147_847_390_008_601_48;
        a[17][6] = 0.191_539_150_378_064_04;
        a[17][7] = 85.699_495_190_512_41;
        a[17][8] = -5.094_095_153_569_584;
        a[17][9] = -1_410.985_179_005_200_5;
        a[17][10] = 2_056.873_083_845_176_6;
        a[17][11] = -726.490_635_953_008_5;
        a[17][12] = 0.0;
        a[17][13] = 5.440_493_103_670_101e-3;
        a[17][14] = -0.143_791_234_650_488_01;
        a[17][15] = 8.130_926_198_406_332e-2;
        a[17][16] = -8.328_707_808_559_681e-2;

        c[18] = 0.451;

        a[18][0] = 4.696_791_108_973_67e-2;
        a[18][1] = 0.0;
        a[18][2] = 0.0;
        a[18][3] = 0.0;
        a[18][4] = 0.0;
        a[18][5] = 0.201_020_760_676_109_74;
        a[18][6] = 0.263_682_085_883_409_74;
        a[18][7] = 129.694_642_865_847_3;
        a[18][8] = -7.848_674_084_672_58;
        a[18][9] = -2_121.729_757_286_215_8;
        a[18][10] = 3_090.157_418_699_412;
        a[18][11] = -1_090.274_673_114_349_5;
        a[18][12] = 0.0;
        a[18][13] = 8.963_759_217_746_588e-3;
        a[18][14] = 4.875_201_082_359_089_5e-2;
        a[18][15] = -2.352_913_684_548_179_8e-2;
        a[18][16] = -9.381_447_086_675_579e-2;
        a[18][17] = 0.0;

        c[19] = 0.705;

        a[19][0] = 4.619_987_481_300_149_4e-2;
        a[19][1] = 0.0;
        a[19][2] = 0.0;
        a[19][3] = 0.0;
        a[19][4] = 0.0;
        a[19][5] = 0.232_283_678_301_755_42;
        a[19][6] = 0.298_205_197_822_691_34;
        a[19][7] = 123.626_170_008_820_07;
        a[19][8] = -7.231_890_372_340_912;
        a[19][9] = -2_046.801_418_638_051_5;
        a[19][10] = 2_986.087_035_469_917_7;
        a[19][11] = -1_055.671_796_398_943;
        a[19][12] = 0.0;
        a[19][13] = 7.930_343_978_313_867e-3;
        a[19][14] = 8.980_932_823_961_332e-2;
        a[19][15] = -8.017_153_994_764_18e-2;
        a[19][16] = 0.102_643_047_389_673_75;
        a[19][17] = 0.0;
        a[19][18] = 0.0;

        c[20] = 0.8795;

        a[20][0] = 5.329_137_345_256_787e-2;
        a[20][1] = 0.0;
        a[20][2] = 0.0;
        a[20][3] = 0.0;
        a[20][4] = 0.0;
        a[20][5] = 0.204_318_580_298_548_3;
        a[20][6] = 0.135_872_889_780_171_29;
        a[20][7] = -402.807_984_460_812_8;
        a[20][8] = 29.461_105_684_176_53;
        a[20][9] = 6_093.592_821_606_032;
        a[20][10] = -8_771.855_718_519_913;
        a[20][11] = 3_051.798_314_820_693;
        a[20][12] = 0.0;
        a[20][13] = -3.308_969_275_605_73e-2;
        a[20][14] = 2.805_382_986_735_396e-2;
        a[20][15] = 0.133_595_363_421_956_27;
        a[20][16] = 0.168_918_525_760_599_15;
        a[20][17] = 0.0;
        a[20][18] = 0.0;
        a[20][19] = 0.0;

        bi8[0][0] = 1.0;
        bi8[0][1] = -8.560_562_317_505_521;
        bi8[0][2] = 37.911_203_197_906_45;
        bi8[0][3] = -95.909_979_703_870_49;
        bi8[0][4] = 143.955_798_956_065_25;
        bi8[0][5] = -126.864_525_883_001_04;
        bi8[0][6] = 60.678_975_081_566_136;
        bi8[0][7] = -12.164_653_899_563_671;

        bi8[1][0] = 0.0;
        bi8[1][1] = 0.0;
        bi8[1][2] = 0.0;
        bi8[1][3] = 0.0;
        bi8[1][4] = 0.0;
        bi8[1][5] = 0.0;
        bi8[1][6] = 0.0;
        bi8[1][7] = 0.0;

        bi8[2][0] = 0.0;
        bi8[2][1] = 0.0;
        bi8[2][2] = 0.0;
        bi8[2][3] = 0.0;
        bi8[2][4] = 0.0;
        bi8[2][5] = 0.0;
        bi8[2][6] = 0.0;
        bi8[2][7] = 0.0;

        bi8[3][0] = 0.0;
        bi8[3][1] = 0.0;
        bi8[3][2] = 0.0;
        bi8[3][3] = 0.0;
        bi8[3][4] = 0.0;
        bi8[3][5] = 0.0;
        bi8[3][6] = 0.0;
        bi8[3][7] = 0.0;

        bi8[4][0] = 0.0;
        bi8[4][1] = 0.0;
        bi8[4][2] = 0.0;
        bi8[4][3] = 0.0;
        bi8[4][4] = 0.0;
        bi8[4][5] = 0.0;
        bi8[4][6] = 0.0;
        bi8[4][7] = 0.0;

        bi8[5][0] = 0.0;
        bi8[5][1] = 31.581_058_769_660_707;
        bi8[5][2] = -180.517_800_649_793_6;
        bi8[5][3] = 371.242_868_417_558_55;
        bi8[5][4] = -219.959_443_261_613_3;
        bi8[5][5] = -240.093_837_615_773_5;
        bi8[5][6] = 371.162_514_909_573_4;
        bi8[5][7] = -133.044_693_953_060_12;

        bi8[6][0] = 0.0;
        bi8[6][1] = 22.070_739_652_927_315;
        bi8[6][2] = -126.156_675_427_491_76;
        bi8[6][3] = 259.446_801_850_776_47;
        bi8[6][4] = -153.720_862_933_629_3;
        bi8[6][5] = -167.791_986_359_261_5;
        bi8[6][6] = 259.390_645_995_842_2;
        bi8[6][7] = -92.979_618_696_708_13;

        bi8[7][0] = 0.0;
        bi8[7][1] = -57_935.131_852_001_3;
        bi8[7][2] = 331_158.073_532_549_84;
        bi8[7][3] = -681_041.275_017_313;
        bi8[7][4] = 403_513.366_679_671_74;
        bi8[7][5] = 440_449.708_813_618_4;
        bi8[7][6] = -680_893.867_322_281_5;
        bi8[7][7] = 244_069.141_018_938_3;

        bi8[8][0] = 0.0;
        bi8[8][1] = 4_250.800_686_968_169;
        bi8[8][2] = -24_297.639_816_601_048;
        bi8[8][3] = 49_969.174_612_265_41;
        bi8[8][4] = -29_606.472_643_654_408;
        bi8[8][5] = -32_316.555_860_831_86;
        bi8[8][6] = 49_958.359_055_084_92;
        bi8[8][7] = -17_907.774_421_940_754;

        bi8[9][0] = 0.0;
        bi8[9][1] = 875_116.529_793_378_3;
        bi8[9][2] = -5_002_179.072_677_031;
        bi8[9][3] = 10_287_203.259_702_617;
        bi8[9][4] = -6_095_113.722_637_695;
        bi8[9][5] = -6_653_041.227_387_194_5;
        bi8[9][6] = 10_284_976.650_278_99;
        bi8[9][7] = -3_686_691.181_850_927_4;

        bi8[10][0] = 0.0;
        bi8[10][1] = -1_259_453.640_882_216_6;
        bi8[10][2] = 7_199_055.703_947_677;
        bi8[10][3] = -14_805_177.549_310_993;
        bi8[10][4] = 8_771_989.681_625_245;
        bi8[10][5] = 9_574_949.976_948_425;
        bi8[10][6] = -14_801_973.048_824_556;
        bi8[10][7] = 5_305_826.679_890_062;

        bi8[11][0] = 0.0;
        bi8[11][1] = 438_049.050_169_491;
        bi8[11][2] = -2_503_894.872_242_029_3;
        bi8[11][3] = 5_149_370.927_637_711;
        bi8[11][4] = -3_050_975.139_855_798;
        bi8[11][5] = -3_330_251.790_676_982_6;
        bi8[11][6] = 5_148_256.374_192_641;
        bi8[11][7] = -1_845_413.171_271_418_2;

        bi8[12][0] = 0.0;
        bi8[12][1] = 0.0;
        bi8[12][2] = 0.0;
        bi8[12][3] = 0.0;
        bi8[12][4] = 0.0;
        bi8[12][5] = 0.0;
        bi8[12][6] = 0.0;
        bi8[12][7] = 0.0;

        bi8[13][0] = 0.0;
        bi8[13][1] = -2.504_431_470_815_825_7;
        bi8[13][2] = 10.020_515_934_447_003;
        bi8[13][3] = 3.188_314_791_667_845;
        bi8[13][4] = -81.289_190_874_560_96;
        bi8[13][5] = 166.501_641_547_533_3;
        bi8[13][6] = -137.964_960_485_395_12;
        bi8[13][7] = 42.048_110_557_123_785;

        bi8[14][0] = 0.0;
        bi8[14][1] = -8.781_714_078_400_347;
        bi8[14][2] = 95.926_836_813_178_82;
        bi8[14][3] = -437.868_401_882_205_94;
        bi8[14][4] = 1_022.893_757_817_680_7;
        bi8[14][5] = -1_279.306_771_651_130_5;
        bi8[14][6] = 814.461_977_782_550_7;
        bi8[14][7] = -207.325_684_801_673_42;

        bi8[15][0] = 0.0;
        bi8[15][1] = 23.683_610_887_782_528;
        bi8[15][2] = -258.707_338_510_044_3;
        bi8[15][3] = 1_180.897_573_941_780_6;
        bi8[15][4] = -2_758.666_192_432_960_5;
        bi8[15][5] = 3_450.192_469_874_827;
        bi8[15][6] = -2_196.541_630_994_369_4;
        bi8[15][7] = 559.141_507_232_983_6;

        bi8[16][0] = 0.0;
        bi8[16][1] = -40.604_955_901_261_88;
        bi8[16][2] = 443.547_232_780_800_47;
        bi8[16][3] = -2_024.619_224_703_986_2;
        bi8[16][4] = 4_729.663_885_325_313;
        bi8[16][5] = -5_915.268_315_880_109;
        bi8[16][6] = 3_765.915_446_103_794_2;
        bi8[16][7] = -958.634_067_724_549_9;

        bi8[17][0] = 0.0;
        bi8[17][1] = -46.254_485_925_330_556;
        bi8[17][2] = 439.193_846_604_121_7;
        bi8[17][3] = -1_719.438_253_681_139_3;
        bi8[17][4] = 3_520.667_063_097_926;
        bi8[17][5] = -3_962.634_989_140_663_5;
        bi8[17][6] = 2_322.579_486_243_480_5;
        bi8[17][7] = -554.112_667_198_394_9;

        bi8[18][0] = 0.0;
        bi8[18][1] = 14.237_752_210_103_935;
        bi8[18][2] = -276.293_272_408_963_9;
        bi8[18][3] = 1_558.151_943_742_051_1;
        bi8[18][4] = -3_984.134_381_318_252;
        bi8[18][5] = 5_195.127_830_395_725;
        bi8[18][6] = -3_374.420_443_020_703_3;
        bi8[18][7] = 867.330_570_400_038_7;

        bi8[19][0] = 0.0;
        bi8[19][1] = 6.177_962_769_210_591;
        bi8[19][2] = -157.491_070_117_552_48;
        bi8[19][3] = 978.638_089_062_525_4;
        bi8[19][4] = -2_692.299_557_325_996;
        bi8[19][5] = 3_734.579_314_715_993;
        bi8[19][6] = -2_556.424_739_731_601;
        bi8[19][7] = 686.820_000_627_420_5;

        bi8[20][0] = 0.0;
        bi8[20][1] = -18.652_890_215_881_367;
        bi8[20][2] = 130.373_777_217_996_12;
        bi8[20][3] = -398.267_356_123_524_5;
        bi8[20][4] = 665.175_955_179_824;
        bi8[20][5] = -644.552_667_038_92;
        bi8[20][6] = 346.695_348_237_773_77;
        bi8[20][7] = -80.772_167_257_267_92;

        let c = c.map(|x| T::from_f64(x).unwrap());
        let a = a.map(|row| row.map(|x| T::from_f64(x).unwrap()));
        let b = b.map(|x| T::from_f64(x).unwrap());
        let bh = bh.map(|x| T::from_f64(x).unwrap());
        let bi8 = bi8.map(|row| row.map(|x| T::from_f64(x).unwrap()));

        Self {
            c,
            a,
            b,
            bh: Some(bh),
            bi: Some(bi8),
            er: None,
        }
    }
}

impl<T: Real> ButcherTableau<T, 16, 21> {
    /// A better efficient Runge-Kutta (16:8(9)) pair with 8th-order interpolation.
    ///
    /// # Overview
    /// This provides a 16-stage Runge-Kutta method with:
    /// - Primary order: 8
    /// - Embedded order: 9 (for error estimation)
    /// - Number of stages: 16 primary + 5 additional stages for interpolation
    /// - Dominant stage order: 5
    ///
    /// # Efficiency
    /// This method is particularly efficient with a local truncation error 2-norm:
    ///
    /// T_10,2 ~ 0.00000034399
    ///
    /// This 2-norm is smaller than previous (16:8,9) pairs and has two local maximum values on [0,1].
    ///
    /// # Interpolation
    /// - Nodes c[17]=1 and c[18]-c[21] were selected to minimize interpolation error.
    /// - The interpolant has continuous derivatives.
    /// - Maximum 2-norm of the local truncation error over [0,1] for the interpolant:
    ///
    ///   Ti_9,2 ~ 0.000003721
    ///
    /// - The interpolation error has three local maximum values on [0,1].
    ///
    /// # Notes
    /// - Exact coefficients of the method are possible using surds in terms of sqrt(6).
    /// - This method shares the same primary stages as the 9th-order method (rkv989e).
    /// - The interpolant requires fewer additional stages than the 9th-order version.
    ///
    /// # Source
    /// [Verner's RKV878e](https://www.sfu.ca/~jverner/RKV98.IIa.Efficient.00000034399.240407.BetterEfficientonWeb)
    pub fn rkv988e() -> Self {
        let mut c = [0.0; 21];
        let mut a = [[0.0; 21]; 21];
        let mut b = [0.0; 16];
        let mut bh = [0.0; 16];
        let mut bi8 = [[0.0; 21]; 21];

        c[0] = 0.0;
        c[1] = 0.3571e-1;
        c[2] = 9.906_028_091_267_415e-2;
        c[3] = 0.148_590_421_369_011_2;
        c[4] = 0.6134;
        c[5] = 0.232_735_947_360_562_7;
        c[6] = 0.553_864_052_639_437_3;
        c[7] = 0.6555;
        c[8] = 0.491625;
        c[9] = 0.6858e-1;
        c[10] = 0.253;
        c[11] = 0.662_064_179_541_204_6;
        c[12] = 0.8309;
        c[13] = 0.8998;
        c[14] = 1.0;
        c[15] = 1.0;

        a[1][0] = 0.3571e-1;

        a[2][0] = -3.833_735_636_677_017e-2;
        a[2][1] = 0.137_397_637_279_444_32;

        a[3][0] = 3.714_760_534_225_28e-2;
        a[3][1] = 0.0;
        a[3][2] = 0.111_442_816_026_758_42;

        a[4][0] = 2.674_764_429_871_505;
        a[4][1] = 0.0;
        a[4][2] = -9.982_382_134_885_293;
        a[4][3] = 7.921_017_705_013_789;

        a[5][0] = 5.242_104_050_577_351e-2;
        a[5][1] = 0.0;
        a[5][2] = 0.0;
        a[5][3] = 0.179_691_118_917_595_32;
        a[5][4] = 6.237_879_371_938_568e-4;

        a[6][0] = 0.159_249_222_364_763_22;
        a[6][1] = 0.0;
        a[6][2] = 0.0;
        a[6][3] = -0.429_842_987_724_108_7;
        a[6][4] = 6.665_266_542_726_088e-2;
        a[6][5] = 0.757_805_152_571_522;

        a[7][0] = 7.283_333_333_333_333e-2;
        a[7][1] = 0.0;
        a[7][2] = 0.0;
        a[7][3] = 0.0;
        a[7][4] = 0.0;
        a[7][5] = 0.335_934_459_066_510_37;
        a[7][6] = 0.246_732_207_600_156_3;

        a[8][0] = 7.297_558_593_75e-2;
        a[8][1] = 0.0;
        a[8][2] = 0.0;
        a[8][3] = 0.0;
        a[8][4] = 0.0;
        a[8][5] = 0.334_800_972_969_933_33;
        a[8][6] = 0.118_415_823_905_066_65;
        a[8][7] = -3.456_738_281_25e-2;

        a[9][0] = 4.911_213_663_452_096_4e-2;
        a[9][1] = 0.0;
        a[9][2] = 0.0;
        a[9][3] = 0.0;
        a[9][4] = 0.0;
        a[9][5] = 3.983_857_361_308_652e-2;
        a[9][6] = 0.106_967_528_893_935_49;
        a[9][7] = -2.174_259_165_458_647_7e-2;
        a[9][8] = -0.105_595_647_486_956_49;

        a[10][0] = -2.707_988_818_641_280_5e-2;
        a[10][1] = 0.0;
        a[10][2] = 0.0;
        a[10][3] = 0.0;
        a[10][4] = 0.0;
        a[10][5] = 3.33e-2;
        a[10][6] = -0.164_552_607_003_605_72;
        a[10][7] = 3.428_266_306_497_39e-2;
        a[10][8] = 0.158_526_406_443_922_1;
        a[10][9] = 0.218_523_425_681_122_5;

        a[11][0] = 5.584_657_769_108_862_5e-2;
        a[11][1] = 0.0;
        a[11][2] = 0.0;
        a[11][3] = 0.0;
        a[11][4] = 0.0;
        a[11][5] = 9.166_533_166_672_539e-2;
        a[11][6] = 0.239_239_965_552_362_7;
        a[11][7] = 1.023_834_712_248_415e-2;
        a[11][8] = -2.679_331_322_859_542_6e-3;
        a[11][9] = 4.235_624_181_474_284_5e-2;
        a[11][10] = 0.225_397_047_016_660_4;

        a[12][0] = -0.480_251_051_272_519_6;
        a[12][1] = 0.0;
        a[12][2] = 0.0;
        a[12][3] = 0.0;
        a[12][4] = 0.0;
        a[12][5] = -6.359_610_162_555_930_5;
        a[12][6] = -0.276_231_389_804_084_1;
        a[12][7] = -6.500_796_633_979_847;
        a[12][8] = 0.573_476_587_704_095_7;
        a[12][9] = 1.347_125_994_868_138_9;
        a[12][10] = 5.936_840_409_706_221;
        a[12][11] = 6.590_346_245_333_925;

        a[13][0] = 0.330_753_306_767_140_1;
        a[13][1] = 0.0;
        a[13][2] = 0.0;
        a[13][3] = 0.0;
        a[13][4] = 0.0;
        a[13][5] = 5.956_207_776_829_962;
        a[13][6] = -0.486_831_640_048_152_77;
        a[13][7] = 4.462_055_288_206_771;
        a[13][8] = 0.741_025_823_144_207_2;
        a[13][9] = -0.711_819_203_457_591_3;
        a[13][10] = -5.454_619_594_516_665;
        a[13][11] = -4.140_803_729_244_71;
        a[13][12] = 0.203_831_972_319_038_66;

        a[14][0] = -0.584_711_112_299_894_5;
        a[14][1] = 0.0;
        a[14][2] = 0.0;
        a[14][3] = 0.0;
        a[14][4] = 0.0;
        a[14][5] = -12.412_684_171_162_67;
        a[14][6] = 1.360_245_445_660_928;
        a[14][7] = -22.426_105_311_118_683;
        a[14][8] = -0.882_885_705_586_545_8;
        a[14][9] = 1.770_155_128_538_230_4;
        a[14][10] = 12.158_096_519_185_339;
        a[14][11] = 22.230_375_204_077_607;
        a[14][12] = -0.663_448_376_020_124_9;
        a[14][13] = 0.450_962_378_725_813_74;

        a[15][0] = 1.940_575_549_810_648_7;
        a[15][1] = 0.0;
        a[15][2] = 0.0;
        a[15][3] = 0.0;
        a[15][4] = 0.0;
        a[15][5] = 21.977_984_081_145_564;
        a[15][6] = 0.823_074_732_698_472_9;
        a[15][7] = 68.164_416_836_263_54;
        a[15][8] = -3.117_097_463_620_267;
        a[15][9] = -4.568_841_021_822_44;
        a[15][10] = -18.741_909_871_262_65;
        a[15][11] = -66.577_118_396_378_32;
        a[15][12] = 1.098_915_553_165_441_8;
        a[15][13] = 0.0;
        a[15][14] = 0.0;

        b[0] = 1.500_669_014_979_724_7e-2;
        b[1] = 0.0;
        b[2] = 0.0;
        b[3] = 0.0;
        b[4] = 0.0;
        b[5] = 0.0;
        b[6] = 0.0;
        b[7] = -1.055_180_992_746_381_3;
        b[8] = 0.238_494_726_378_218_3;
        b[9] = 0.128_815_177_428_299_15;
        b[10] = 0.227_662_311_104_621_57;
        b[11] = 1.229_532_587_437_517_4;
        b[12] = 4.624_976_662_810_384e-2;
        b[13] = 0.138_619_631_936_629_38;
        b[14] = 3.080_010_168_319_435_5e-2;
        b[15] = 0.0;

        bh[0] = 1.897_210_532_481_101_4e-2;
        bh[1] = 0.0;
        bh[2] = 0.0;
        bh[3] = 0.0;
        bh[4] = 0.0;
        bh[5] = 0.0;
        bh[6] = 0.0;
        bh[7] = 3.408_110_314_549_493_8;
        bh[8] = 0.126_032_388_382_092_1;
        bh[9] = 0.118_837_506_345_114_97;
        bh[10] = 0.249_104_199_783_868_75;
        bh[11] = -3.269_966_219_928_978_3;
        bh[12] = 0.302_379_810_022_888_3;
        bh[13] = 0.0;
        bh[14] = 0.0;
        bh[15] = 4.652_989_552_070_924e-2;

        c[16] = 1.0;

        a[16][0] = 1.500_669_014_979_724_7e-2;
        a[16][1] = 0.0;
        a[16][2] = 0.0;
        a[16][3] = 0.0;
        a[16][4] = 0.0;
        a[16][5] = 0.0;
        a[16][6] = 0.0;
        a[16][7] = -1.055_180_992_746_381_3;
        a[16][8] = 0.238_494_726_378_218_3;
        a[16][9] = 0.128_815_177_428_299_15;
        a[16][10] = 0.227_662_311_104_621_57;
        a[16][11] = 1.229_532_587_437_517_4;
        a[16][12] = 4.624_976_662_810_384e-2;
        a[16][13] = 0.138_619_631_936_629_38;
        a[16][14] = 3.080_010_168_319_435_5e-2;
        a[16][15] = 0.0;

        c[17] = 0.737_501_813_998_881;

        a[17][0] = 1.571_801_061_417_788e-2;
        a[17][1] = 0.0;
        a[17][2] = 0.0;
        a[17][3] = 0.0;
        a[17][4] = 0.0;
        a[17][5] = 0.0;
        a[17][6] = 0.0;
        a[17][7] = 0.485_340_345_265_736_33;
        a[17][8] = 0.210_778_756_890_454_67;
        a[17][9] = 0.126_980_241_305_335_42;
        a[17][10] = 0.231_968_701_451_391_92;
        a[17][11] = -0.362_021_471_406_909_66;
        a[17][12] = 5.366_106_712_036_344e-2;
        a[17][13] = -2.806_066_613_385_549_8e-2;
        a[17][14] = -2.378_121_372_710_33e-2;
        a[17][15] = 0.0;
        a[17][16] = 2.691_804_261_928_989e-2;

        c[18] = 0.749;

        a[18][0] = 1.569_705_832_522_204_4e-2;
        a[18][1] = 0.0;
        a[18][2] = 0.0;
        a[18][3] = 0.0;
        a[18][4] = 0.0;
        a[18][5] = 0.0;
        a[18][6] = 0.0;
        a[18][7] = 0.461_607_524_220_211_2;
        a[18][8] = 0.211_394_651_669_811_33;
        a[18][9] = 0.127_033_091_716_710_93;
        a[18][10] = 0.231_854_055_029_870_83;
        a[18][11] = -0.338_526_640_668_837_3;
        a[18][12] = 5.298_251_972_194_236e-2;
        a[18][13] = -2.750_461_365_887_187_8e-2;
        a[18][14] = -2.361_906_185_395_527e-2;
        a[18][15] = 0.0;
        a[18][16] = 2.668_458_089_504_036e-2;
        a[18][17] = 1.139_683_460_285_541_5e-2;

        c[19] = 0.65;

        a[19][0] = 1.438_964_884_291_216_3e-2;
        a[19][1] = 0.0;
        a[19][2] = 0.0;
        a[19][3] = 0.0;
        a[19][4] = 0.0;
        a[19][5] = 0.0;
        a[19][6] = 0.0;
        a[19][7] = -1.206_901_219_123_788_4;
        a[19][8] = 0.250_562_855_463_937_64;
        a[19][9] = 0.130_333_291_570_212_73;
        a[19][10] = 0.224_671_775_926_352_2;
        a[19][11] = 1.308_419_325_781_946_5;
        a[19][12] = 2.589_750_180_376_236e-3;
        a[19][13] = 8.070_743_254_562_857e-3;
        a[19][14] = -1.267_568_255_392_829_4e-2;
        a[19][15] = 0.0;
        a[19][16] = 1.129_158_072_373_321_6e-2;
        a[19][17] = 3.422_056_680_709_75e-2;
        a[19][18] = -0.114_972_636_873_414_2;

        c[20] = 0.487;

        a[20][0] = 1.452_348_029_801_042e-2;
        a[20][1] = 0.0;
        a[20][2] = 0.0;
        a[20][3] = 0.0;
        a[20][4] = 0.0;
        a[20][5] = 0.0;
        a[20][6] = 0.0;
        a[20][7] = -0.521_424_310_246_581_9;
        a[20][8] = 0.186_669_884_420_460_4;
        a[20][9] = 0.129_931_635_445_127_3;
        a[20][10] = 0.226_214_108_576_571_93;
        a[20][11] = 0.610_458_263_946_671;
        a[20][12] = 1.418_715_607_022_412_5e-2;
        a[20][13] = 1.480_061_054_412_245_8e-2;
        a[20][14] = -3.711_471_609_871_774_7e-3;
        a[20][15] = 0.0;
        a[20][16] = 1.393_256_979_572_559e-3;
        a[20][17] = 1.114_731_020_624_073_3;
        a[20][18] = -1.021_208_555_757_145_8;
        a[20][19] = -0.279_565_079_291_234_1;

        bi8[0][0] = 1.0;
        bi8[0][1] = -11.009_172_273_099_548;
        bi8[0][2] = 53.603_768_441_854_456;
        bi8[0][3] = -143.781_860_105_177_34;
        bi8[0][4] = 227.780_227_272_405_97;
        bi8[0][5] = -212.766_777_474_539_77;
        bi8[0][6] = 108.476_558_503_094_35;
        bi8[0][7] = -23.287_737_674_388_335;

        bi8[1][0] = 0.0;
        bi8[1][1] = 0.0;
        bi8[1][2] = 0.0;
        bi8[1][3] = 0.0;
        bi8[1][4] = 0.0;
        bi8[1][5] = 0.0;
        bi8[1][6] = 0.0;
        bi8[1][7] = 0.0;

        bi8[2][0] = 0.0;
        bi8[2][1] = 0.0;
        bi8[2][2] = 0.0;
        bi8[2][3] = 0.0;
        bi8[2][4] = 0.0;
        bi8[2][5] = 0.0;
        bi8[2][6] = 0.0;
        bi8[2][7] = 0.0;

        bi8[3][0] = 0.0;
        bi8[3][1] = 0.0;
        bi8[3][2] = 0.0;
        bi8[3][3] = 0.0;
        bi8[3][4] = 0.0;
        bi8[3][5] = 0.0;
        bi8[3][6] = 0.0;
        bi8[3][7] = 0.0;

        bi8[4][0] = 0.0;
        bi8[4][1] = 0.0;
        bi8[4][2] = 0.0;
        bi8[4][3] = 0.0;
        bi8[4][4] = 0.0;
        bi8[4][5] = 0.0;
        bi8[4][6] = 0.0;
        bi8[4][7] = 0.0;

        bi8[5][0] = 0.0;
        bi8[5][1] = 0.0;
        bi8[5][2] = 0.0;
        bi8[5][3] = 0.0;
        bi8[5][4] = 0.0;
        bi8[5][5] = 0.0;
        bi8[5][6] = 0.0;
        bi8[5][7] = 0.0;

        bi8[6][0] = 0.0;
        bi8[6][1] = 0.0;
        bi8[6][2] = 0.0;
        bi8[6][3] = 0.0;
        bi8[6][4] = 0.0;
        bi8[6][5] = 0.0;
        bi8[6][6] = 0.0;
        bi8[6][7] = 0.0;

        bi8[7][0] = 0.0;
        bi8[7][1] = 179.087_906_217_887_22;
        bi8[7][2] = -1_728.403_136_057_631_4;
        bi8[7][3] = 6_564.460_940_561_143;
        bi8[7][4] = -12_902.910_815_841_211;
        bi8[7][5] = 13_965.179_771_002_651;
        bi8[7][6] = -7_920.424_061_687_376;
        bi8[7][7] = 1_841.954_214_811_791_2;

        bi8[8][0] = 0.0;
        bi8[8][1] = -22.883_047_147_673_906;
        bi8[8][2] = 238.804_050_768_529_4;
        bi8[8][3] = -936.858_578_229_607_4;
        bi8[8][4] = 1_874.044_957_762_151_4;
        bi8[8][5] = -2_050.511_559_353_009_5;
        bi8[8][6] = 1_171.508_545_192_416_2;
        bi8[8][7] = -273.865_874_266_428_2;

        bi8[9][0] = 0.0;
        bi8[9][1] = 12.173_315_496_281_248;
        bi8[9][2] = -82.750_764_764_368_7;
        bi8[9][3] = 256.480_443_875_130_8;
        bi8[9][4] = -441.099_942_975_831_1;
        bi8[9][5] = 434.507_983_140_121_3;
        bi8[9][6] = -229.893_460_589_690_02;
        bi8[9][7] = 50.711_240_995_784_785;

        bi8[10][0] = 0.0;
        bi8[10][1] = 2.613_203_520_133_623;
        bi8[10][2] = 16.880_029_372_611_844;
        bi8[10][3] = -134.172_792_115_523_78;
        bi8[10][4] = 340.120_480_169_263_8;
        bi8[10][5] = -420.125_889_377_687_94;
        bi8[10][6] = 258.323_437_214_655_7;
        bi8[10][7] = -63.410_806_472_348_604;

        bi8[11][0] = 0.0;
        bi8[11][1] = -212.312_700_591_366_6;
        bi8[11][2] = 2_045.352_316_395_154_3;
        bi8[11][3] = -7_762.060_413_644_091;
        bi8[11][4] = 15_250.148_414_593_781;
        bi8[11][5] = -16_501.080_035_445_18;
        bi8[11][6] = 9_356.907_363_957_305;
        bi8[11][7] = -2_175.725_412_678_167;

        bi8[12][0] = 0.0;
        bi8[12][1] = -11.501_647_085_674_298;
        bi8[12][2] = 107.276_970_404_572_52;
        bi8[12][3] = -401.234_533_298_394_75;
        bi8[12][4] = 781.891_698_957_293;
        bi8[12][5] = -841.659_337_101_325_5;
        bi8[12][6] = 475.576_739_148_558_83;
        bi8[12][7] = -110.303_641_258_401_8;

        bi8[13][0] = 0.0;
        bi8[13][1] = -38.772_396_773_403_656;
        bi8[13][2] = 358.639_118_015_297;
        bi8[13][3] = -1_336.215_732_140_227;
        bi8[13][4] = 2_598.193_515_313_187;
        bi8[13][5] = -2_792.878_330_051_016_7;
        bi8[13][6] = 1_576.586_790_342_81;
        bi8[13][7] = -365.414_345_074_710_1;

        bi8[14][0] = 0.0;
        bi8[14][1] = -10.004_254_699_988_527;
        bi8[14][2] = 91.677_542_415_604_98;
        bi8[14][3] = -340.077_760_065_284_85;
        bi8[14][4] = 659.600_836_997_697_8;
        bi8[14][5] = -707.882_628_411_297_6;
        bi8[14][6] = 399.158_003_026_012_7;
        bi8[14][7] = -92.440_939_161_061_42;

        bi8[15][0] = 0.0;
        bi8[15][1] = 0.0;
        bi8[15][2] = 0.0;
        bi8[15][3] = 0.0;
        bi8[15][4] = 0.0;
        bi8[15][5] = 0.0;
        bi8[15][6] = 0.0;
        bi8[15][7] = 0.0;

        bi8[16][0] = 0.0;
        bi8[16][1] = 16.210_144_913_375_61;
        bi8[16][2] = -150.449_213_104_866_3;
        bi8[16][3] = 565.119_663_172_215_3;
        bi8[16][4] = -1_111.210_282_651_553_6;
        bi8[16][5] = 1_211.236_976_000_843_5;
        bi8[16][6] = -695.336_560_691_809_2;
        bi8[16][7] = 164.429_272_361_794_77;

        bi8[17][0] = 0.0;
        bi8[17][1] = -1_872.805_019_318_415;
        bi8[17][2] = 16_400.666_332_070_487;
        bi8[17][3] = -57_270.162_601_660_53;
        bi8[17][4] = 103_330.591_296_066_36;
        bi8[17][5] = -102_051.188_817_372_75;
        bi8[17][6] = 52_424.752_608_746_625;
        bi8[17][7] = -10_961.853_798_531_77;

        bi8[18][0] = 0.0;
        bi8[18][1] = 1_720.562_129_057_968_8;
        bi8[18][2] = -15_159.620_464_322_548;
        bi8[18][3] = 53_314.595_619_355_234;
        bi8[18][4] = -96_988.374_686_915_76;
        bi8[18][5] = 96_690.391_810_236_88;
        bi8[18][6] = -50_199.312_489_882_47;
        bi8[18][7] = 10_621.758_082_470_702;

        bi8[19][0] = 0.0;
        bi8[19][1] = 264.247_048_839_608_45;
        bi8[19][2] = -2_232.338_003_905_684_7;
        bi8[19][3] = 7_453.753_802_419_144;
        bi8[19][4] = -12_770.718_814_009_168;
        bi8[19][5] = 11_903.252_267_674_463;
        bi8[19][6] = -5_733.155_576_507_225;
        bi8[19][7] = 1_114.959_275_488_863;

        bi8[20][0] = 0.0;
        bi8[20][1] = -15.605_510_155_633_654;
        bi8[20][2] = 40.661_454_270_987_6;
        bi8[20][3] = 170.153_801_875_971_4;
        bi8[20][4] = -848.056_884_738_605_3;
        bi8[20][5] = 1_373.524_566_531_853_5;
        bi8[20][6] = -993.167_896_772_913;
        bi8[20][7] = 272.490_468_988_339_4;

        let c = c.map(|x| T::from_f64(x).unwrap());
        let a = a.map(|row| row.map(|x| T::from_f64(x).unwrap()));
        let b = b.map(|x| T::from_f64(x).unwrap());
        let bh = bh.map(|x| T::from_f64(x).unwrap());
        let bi8 = bi8.map(|row| row.map(|x| T::from_f64(x).unwrap()));

        Self {
            c,
            a,
            b,
            bh: Some(bh),
            bi: Some(bi8),
            er: None,
        }
    }
}

impl<T: Real> ButcherTableau<T, 16, 26> {
    /// A better efficient Runge-Kutta (16:9(8)) pair with 9th-order interpolation.
    ///
    /// # Overview
    /// This provides a 16-stage Runge-Kutta method with:
    /// - Primary order: 9
    /// - Embedded order: 8 (for error estimation)
    /// - Number of stages: 16 primary + 10 additional stages for interpolation
    /// - Dominant stage order: 5
    ///
    /// # Efficiency
    /// This method is particularly efficient with a local truncation error 2-norm:
    ///
    /// T_10,2 ~ 0.00000034399
    ///
    /// This 2-norm is smaller than previous (16:8,9) pairs and has two local maximum values on [0,1].
    ///
    /// # Interpolation
    /// - The method provides a 9th-order interpolant using additional nodes c[17] through c[25].
    /// - The interpolant has continuous derivatives.
    /// - Maximum 2-norm of the local truncation error over [0,1] for the interpolant:
    ///
    ///   Ti_10,2 ~ 0.0000007830
    ///
    /// - The interpolation error has two local maximum values on [0,1].
    ///
    /// # Notes
    /// - Exact coefficients of the method are possible using surds in terms of sqrt(6).
    /// - Exact coefficients for the interpolants cannot be conveniently represented
    ///   as they require c[18] to be a zero of a cubic polynomial with coefficients
    ///   that are surds in terms of sqrt(6).
    ///
    /// # Source
    /// [Verner's RKV878e](https://www.sfu.ca/~jverner/RKV98.IIa.Efficient.00000034399.240407.BetterEfficientonWeb)
    pub fn rkv989e() -> Self {
        let mut c = [0.0; 26];
        let mut a = [[0.0; 26]; 26];
        let mut b = [0.0; 16];
        let mut bh = [0.0; 16];
        let mut bi9 = [[0.0; 26]; 26];

        c[0] = 0.0;
        c[1] = 0.3571e-1;
        c[2] = 9.906_028_091_267_415e-2;
        c[3] = 0.148_590_421_369_011_2;
        c[4] = 0.6134;
        c[5] = 0.232_735_947_360_562_7;
        c[6] = 0.553_864_052_639_437_3;
        c[7] = 0.6555;
        c[8] = 0.491625;
        c[9] = 0.6858e-1;
        c[10] = 0.253;
        c[11] = 0.662_064_179_541_204_6;
        c[12] = 0.8309;
        c[13] = 0.8998;
        c[14] = 1.0;
        c[15] = 1.0;

        a[1][0] = 0.3571e-1;

        a[2][0] = -3.833_735_636_677_017e-2;
        a[2][1] = 0.137_397_637_279_444_32;

        a[3][0] = 3.714_760_534_225_28e-2;
        a[3][1] = 0.0;
        a[3][2] = 0.111_442_816_026_758_42;

        a[4][0] = 2.674_764_429_871_505;
        a[4][1] = 0.0;
        a[4][2] = -9.982_382_134_885_293;
        a[4][3] = 7.921_017_705_013_789;

        a[5][0] = 5.242_104_050_577_351e-2;
        a[5][1] = 0.0;
        a[5][2] = 0.0;
        a[5][3] = 0.179_691_118_917_595_32;
        a[5][4] = 6.237_879_371_938_568e-4;

        a[6][0] = 0.159_249_222_364_763_22;
        a[6][1] = 0.0;
        a[6][2] = 0.0;
        a[6][3] = -0.429_842_987_724_108_7;
        a[6][4] = 6.665_266_542_726_088e-2;
        a[6][5] = 0.757_805_152_571_522;

        a[7][0] = 7.283_333_333_333_333e-2;
        a[7][1] = 0.0;
        a[7][2] = 0.0;
        a[7][3] = 0.0;
        a[7][4] = 0.0;
        a[7][5] = 0.335_934_459_066_510_37;
        a[7][6] = 0.246_732_207_600_156_3;

        a[8][0] = 7.297_558_593_75e-2;
        a[8][1] = 0.0;
        a[8][2] = 0.0;
        a[8][3] = 0.0;
        a[8][4] = 0.0;
        a[8][5] = 0.334_800_972_969_933_33;
        a[8][6] = 0.118_415_823_905_066_65;
        a[8][7] = -3.456_738_281_25e-2;

        a[9][0] = 4.911_213_663_452_096_4e-2;
        a[9][1] = 0.0;
        a[9][2] = 0.0;
        a[9][3] = 0.0;
        a[9][4] = 0.0;
        a[9][5] = 3.983_857_361_308_652e-2;
        a[9][6] = 0.106_967_528_893_935_49;
        a[9][7] = -2.174_259_165_458_647_7e-2;
        a[9][8] = -0.105_595_647_486_956_49;

        a[10][0] = -2.707_988_818_641_280_5e-2;
        a[10][1] = 0.0;
        a[10][2] = 0.0;
        a[10][3] = 0.0;
        a[10][4] = 0.0;
        a[10][5] = 3.33e-2;
        a[10][6] = -0.164_552_607_003_605_72;
        a[10][7] = 3.428_266_306_497_39e-2;
        a[10][8] = 0.158_526_406_443_922_1;
        a[10][9] = 0.218_523_425_681_122_5;

        a[11][0] = 5.584_657_769_108_862_5e-2;
        a[11][1] = 0.0;
        a[11][2] = 0.0;
        a[11][3] = 0.0;
        a[11][4] = 0.0;
        a[11][5] = 9.166_533_166_672_539e-2;
        a[11][6] = 0.239_239_965_552_362_7;
        a[11][7] = 1.023_834_712_248_415e-2;
        a[11][8] = -2.679_331_322_859_542_6e-3;
        a[11][9] = 4.235_624_181_474_284_5e-2;
        a[11][10] = 0.225_397_047_016_660_4;

        a[12][0] = -0.480_251_051_272_519_6;
        a[12][1] = 0.0;
        a[12][2] = 0.0;
        a[12][3] = 0.0;
        a[12][4] = 0.0;
        a[12][5] = -6.359_610_162_555_930_5;
        a[12][6] = -0.276_231_389_804_084_1;
        a[12][7] = -6.500_796_633_979_847;
        a[12][8] = 0.573_476_587_704_095_7;
        a[12][9] = 1.347_125_994_868_138_9;
        a[12][10] = 5.936_840_409_706_221;
        a[12][11] = 6.590_346_245_333_925;

        a[13][0] = 0.330_753_306_767_140_1;
        a[13][1] = 0.0;
        a[13][2] = 0.0;
        a[13][3] = 0.0;
        a[13][4] = 0.0;
        a[13][5] = 5.956_207_776_829_962;
        a[13][6] = -0.486_831_640_048_152_77;
        a[13][7] = 4.462_055_288_206_771;
        a[13][8] = 0.741_025_823_144_207_2;
        a[13][9] = -0.711_819_203_457_591_3;
        a[13][10] = -5.454_619_594_516_665;
        a[13][11] = -4.140_803_729_244_71;
        a[13][12] = 0.203_831_972_319_038_66;

        a[14][0] = -0.584_711_112_299_894_5;
        a[14][1] = 0.0;
        a[14][2] = 0.0;
        a[14][3] = 0.0;
        a[14][4] = 0.0;
        a[14][5] = -12.412_684_171_162_67;
        a[14][6] = 1.360_245_445_660_928;
        a[14][7] = -22.426_105_311_118_683;
        a[14][8] = -0.882_885_705_586_545_8;
        a[14][9] = 1.770_155_128_538_230_4;
        a[14][10] = 12.158_096_519_185_339;
        a[14][11] = 22.230_375_204_077_607;
        a[14][12] = -0.663_448_376_020_124_9;
        a[14][13] = 0.450_962_378_725_813_74;

        a[15][0] = 1.940_575_549_810_648_7;
        a[15][1] = 0.0;
        a[15][2] = 0.0;
        a[15][3] = 0.0;
        a[15][4] = 0.0;
        a[15][5] = 21.977_984_081_145_564;
        a[15][6] = 0.823_074_732_698_472_9;
        a[15][7] = 68.164_416_836_263_54;
        a[15][8] = -3.117_097_463_620_267;
        a[15][9] = -4.568_841_021_822_44;
        a[15][10] = -18.741_909_871_262_65;
        a[15][11] = -66.577_118_396_378_32;
        a[15][12] = 1.098_915_553_165_441_8;
        a[15][13] = 0.0;
        a[15][14] = 0.0;

        b[0] = 1.500_669_014_979_724_7e-2;
        b[1] = 0.0;
        b[2] = 0.0;
        b[3] = 0.0;
        b[4] = 0.0;
        b[5] = 0.0;
        b[6] = 0.0;
        b[7] = -1.055_180_992_746_381_3;
        b[8] = 0.238_494_726_378_218_3;
        b[9] = 0.128_815_177_428_299_15;
        b[10] = 0.227_662_311_104_621_57;
        b[11] = 1.229_532_587_437_517_4;
        b[12] = 4.624_976_662_810_384e-2;
        b[13] = 0.138_619_631_936_629_38;
        b[14] = 3.080_010_168_319_435_5e-2;
        b[15] = 0.0;

        bh[0] = 1.897_210_532_481_101_4e-2;
        bh[1] = 0.0;
        bh[2] = 0.0;
        bh[3] = 0.0;
        bh[4] = 0.0;
        bh[5] = 0.0;
        bh[6] = 0.0;
        bh[7] = 3.408_110_314_549_493_8;
        bh[8] = 0.126_032_388_382_092_1;
        bh[9] = 0.118_837_506_345_114_97;
        bh[10] = 0.249_104_199_783_868_75;
        bh[11] = -3.269_966_219_928_978_3;
        bh[12] = 0.302_379_810_022_888_3;
        bh[13] = 0.0;
        bh[14] = 0.0;
        bh[15] = 4.652_989_552_070_924e-2;

        c[16] = 1.0;

        a[16][0] = 1.500_669_014_979_724_7e-2;
        a[16][1] = 0.0;
        a[16][2] = 0.0;
        a[16][3] = 0.0;
        a[16][4] = 0.0;
        a[16][5] = 0.0;
        a[16][6] = 0.0;
        a[16][7] = -1.055_180_992_746_381_3;
        a[16][8] = 0.238_494_726_378_218_3;
        a[16][9] = 0.128_815_177_428_299_15;
        a[16][10] = 0.227_662_311_104_621_57;
        a[16][11] = 1.229_532_587_437_517_4;
        a[16][12] = 4.624_976_662_810_384e-2;
        a[16][13] = 0.138_619_631_936_629_38;
        a[16][14] = 3.080_010_168_319_435_5e-2;
        a[16][15] = 0.0;

        c[17] = 0.737_501_813_998_881;

        a[17][0] = 1.571_801_061_417_788e-2;
        a[17][1] = 0.0;
        a[17][2] = 0.0;
        a[17][3] = 0.0;
        a[17][4] = 0.0;
        a[17][5] = 0.0;
        a[17][6] = 0.0;
        a[17][7] = 0.485_340_345_265_736_33;
        a[17][8] = 0.210_778_756_890_454_67;
        a[17][9] = 0.126_980_241_305_335_42;
        a[17][10] = 0.231_968_701_451_391_92;
        a[17][11] = -0.362_021_471_406_909_66;
        a[17][12] = 5.366_106_712_036_344e-2;
        a[17][13] = -2.806_066_613_385_549_8e-2;
        a[17][14] = -2.378_121_372_710_33e-2;
        a[17][15] = 0.0;
        a[17][16] = 2.691_804_261_928_989e-2;

        c[18] = 0.749;

        a[18][0] = 1.569_705_832_522_204_4e-2;
        a[18][1] = 0.0;
        a[18][2] = 0.0;
        a[18][3] = 0.0;
        a[18][4] = 0.0;
        a[18][5] = 0.0;
        a[18][6] = 0.0;
        a[18][7] = 0.461_607_524_220_211_2;
        a[18][8] = 0.211_394_651_669_811_33;
        a[18][9] = 0.127_033_091_716_710_93;
        a[18][10] = 0.231_854_055_029_870_83;
        a[18][11] = -0.338_526_640_668_837_3;
        a[18][12] = 5.298_251_972_194_236e-2;
        a[18][13] = -2.750_461_365_887_187_8e-2;
        a[18][14] = -2.361_906_185_395_527e-2;
        a[18][15] = 0.0;
        a[18][16] = 2.668_458_089_504_036e-2;
        a[18][17] = 1.139_683_460_285_541_5e-2;

        c[19] = 0.65;

        a[19][0] = 1.438_964_884_291_216_3e-2;
        a[19][1] = 0.0;
        a[19][2] = 0.0;
        a[19][3] = 0.0;
        a[19][4] = 0.0;
        a[19][5] = 0.0;
        a[19][6] = 0.0;
        a[19][7] = -1.206_901_219_123_788_4;
        a[19][8] = 0.250_562_855_463_937_64;
        a[19][9] = 0.130_333_291_570_212_73;
        a[19][10] = 0.224_671_775_926_352_2;
        a[19][11] = 1.308_419_325_781_946_5;
        a[19][12] = 2.589_750_180_376_236e-3;
        a[19][13] = 8.070_743_254_562_857e-3;
        a[19][14] = -1.267_568_255_392_829_4e-2;
        a[19][15] = 0.0;
        a[19][16] = 1.129_158_072_373_321_6e-2;
        a[19][17] = 3.422_056_680_709_75e-2;
        a[19][18] = -0.114_972_636_873_414_2;

        c[20] = 0.487;

        a[20][0] = 1.452_348_029_801_042e-2;
        a[20][1] = 0.0;
        a[20][2] = 0.0;
        a[20][3] = 0.0;
        a[20][4] = 0.0;
        a[20][5] = 0.0;
        a[20][6] = 0.0;
        a[20][7] = -0.521_424_310_246_581_9;
        a[20][8] = 0.186_669_884_420_460_4;
        a[20][9] = 0.129_931_635_445_127_3;
        a[20][10] = 0.226_214_108_576_571_93;
        a[20][11] = 0.610_458_263_946_671;
        a[20][12] = 1.418_715_607_022_412_5e-2;
        a[20][13] = 1.480_061_054_412_245_8e-2;
        a[20][14] = -3.711_471_609_871_774_7e-3;
        a[20][15] = 0.0;
        a[20][16] = 1.393_256_979_572_559e-3;
        a[20][17] = 1.114_731_020_624_073_3;
        a[20][18] = -1.021_208_555_757_145_8;
        a[20][19] = -0.279_565_079_291_234_1;

        c[21] = 0.97e-2;

        a[21][0] = 8.711_816_186_418_633e-3;
        a[21][1] = 0.0;
        a[21][2] = 0.0;
        a[21][3] = 0.0;
        a[21][4] = 0.0;
        a[21][5] = 0.0;
        a[21][6] = 0.0;
        a[21][7] = 1.532_993_247_326_560_3e-2;
        a[21][8] = -1.943_250_606_288_015_1e-3;
        a[21][9] = 1.072_095_950_570_478_4e-3;
        a[21][10] = 2.601_233_036_074_381e-4;
        a[21][11] = -1.817_718_521_410_219_3e-2;
        a[21][12] = -9.877_668_338_996_713e-4;
        a[21][13] = -3.332_383_192_417_756_6e-3;
        a[21][14] = -8.605_833_352_714_281e-4;
        a[21][15] = 0.0;
        a[21][16] = 1.392_810_143_886_650_6e-3;
        a[21][17] = -0.161_741_998_789_763;
        a[21][18] = 0.148_515_656_130_604_4;
        a[21][19] = 2.289_051_095_253_062e-2;
        a[21][20] = -1.429_777_169_141_779_3e-3;

        c[22] = 0.138;

        a[22][0] = 2.710_092_628_714_179_4e-2;
        a[22][1] = 0.0;
        a[22][2] = 0.0;
        a[22][3] = 0.0;
        a[22][4] = 0.0;
        a[22][5] = 0.0;
        a[22][6] = 0.0;
        a[22][7] = 0.692_310_498_600_388_5;
        a[22][8] = -6.725_243_132_164_495e-2;
        a[22][9] = 8.808_440_659_269_46e-2;
        a[22][10] = 5.982_566_312_199_631e-2;
        a[22][11] = -0.825_129_031_481_552_4;
        a[22][12] = -4.886_453_387_508_305_6e-2;
        a[22][13] = -0.168_260_837_156_651_2;
        a[22][14] = -4.443_170_503_743_608_4e-2;
        a[22][15] = 0.0;
        a[22][16] = 7.037_844_639_432_78e-2;
        a[22][17] = -8.818_686_397_504_859;
        a[22][18] = 8.028_821_919_603_92;
        a[22][19] = 1.306_591_406_499_158;
        a[22][20] = -0.162_488_330_722_401_46;
        a[22][21] = 0.0;

        c[23] = 0.249;

        a[23][0] = 1.466_442_426_857_961_2e-2;
        a[23][1] = 0.0;
        a[23][2] = 0.0;
        a[23][3] = 0.0;
        a[23][4] = 0.0;
        a[23][5] = 0.0;
        a[23][6] = 0.0;
        a[23][7] = 0.189_839_956_139_912_45;
        a[23][8] = 3.711_530_651_907_537_4e-2;
        a[23][9] = 0.131_624_798_119_951_04;
        a[23][10] = 0.146_662_292_882_268_56;
        a[23][11] = -0.237_733_116_246_783_84;
        a[23][12] = -2.493_071_232_192_949_8e-2;
        a[23][13] = -9.427_777_049_221_307e-2;
        a[23][14] = -2.726_667_627_690_941_2e-2;
        a[23][15] = 0.0;
        a[23][16] = 4.097_659_404_063_362e-2;
        a[23][17] = -5.538_725_321_977_495;
        a[23][18] = 4.973_442_175_247_713;
        a[23][19] = 0.862_629_444_918_985_9;
        a[23][20] = -0.225_021_394_821_789_32;
        a[23][21] = 0.0;
        a[23][22] = 0.0;

        c[24] = 0.439;

        a[24][0] = 1.191_252_689_920_92e-2;
        a[24][1] = 0.0;
        a[24][2] = 0.0;
        a[24][3] = 0.0;
        a[24][4] = 0.0;
        a[24][5] = 0.0;
        a[24][6] = 0.0;
        a[24][7] = -0.672_313_785_813_087_6;
        a[24][8] = 0.180_128_426_682_568_9;
        a[24][9] = 0.136_568_944_744_934_72;
        a[24][10] = 0.211_103_839_379_890_66;
        a[24][11] = 0.777_585_472_624_472_9;
        a[24][12] = 2.362_117_948_505_939_8e-2;
        a[24][13] = 6.391_325_607_581_23e-2;
        a[24][14] = 1.197_654_289_246_164_4e-2;
        a[24][15] = 0.0;
        a[24][16] = -2.188_622_145_087_031_8e-2;
        a[24][17] = 3.500_218_060_218_659;
        a[24][18] = -3.195_765_244_251_373_5;
        a[24][19] = -0.599_440_104_860_014_9;
        a[24][20] = 1.137_710_737_227_729_3e-2;
        a[24][21] = 0.0;
        a[24][22] = 0.0;
        a[24][23] = 0.0;

        c[25] = 0.794;

        a[25][0] = 1.184_012_014_074_604_2e-2;
        a[25][1] = 0.0;
        a[25][2] = 0.0;
        a[25][3] = 0.0;
        a[25][4] = 0.0;
        a[25][5] = 0.0;
        a[25][6] = 0.0;
        a[25][7] = -0.664_126_328_976_879_1;
        a[25][8] = 0.178_891_406_187_319_4;
        a[25][9] = 0.136_756_264_228_708_5;
        a[25][10] = 0.210_775_811_973_028_6;
        a[25][11] = 0.767_918_474_480_633_6;
        a[25][12] = 2.313_496_869_580_297_3e-2;
        a[25][13] = 6.230_605_705_999_707e-2;
        a[25][14] = 1.157_099_029_193_866_4e-2;
        a[25][15] = 0.0;
        a[25][16] = -2.129_441_601_042_184_4e-2;
        a[25][17] = 3.163_166_251_075_345_3;
        a[25][18] = -2.819_544_872_276_817;
        a[25][19] = -0.397_876_904_630_975_86;
        a[25][20] = 0.130_482_177_761_573_7;
        a[25][21] = 0.0;
        a[25][22] = 0.0;
        a[25][23] = 0.0;
        a[25][24] = 0.0;

        bi9[0][0] = 1.0;
        bi9[0][1] = -60.671_564_990_962_76;
        bi9[0][2] = 669.417_333_989_096_5;
        bi9[0][3] = -3_377.878_946_225_199_3;
        bi9[0][4] = 9_286.468_967_391_047;
        bi9[0][5] = -14_780.477_136_811_021;
        bi9[0][6] = 13_604.993_863_282_247;
        bi9[0][7] = -6_724.705_443_356_114;
        bi9[0][8] = 1_381.867_933_411_056_6;

        bi9[1][0] = 0.0;
        bi9[1][1] = 0.0;
        bi9[1][2] = 0.0;
        bi9[1][3] = 0.0;
        bi9[1][4] = 0.0;
        bi9[1][5] = 0.0;
        bi9[1][6] = 0.0;
        bi9[1][7] = 0.0;
        bi9[1][8] = 0.0;

        bi9[2][0] = 0.0;
        bi9[2][1] = 0.0;
        bi9[2][2] = 0.0;
        bi9[2][3] = 0.0;
        bi9[2][4] = 0.0;
        bi9[2][5] = 0.0;
        bi9[2][6] = 0.0;
        bi9[2][7] = 0.0;
        bi9[2][8] = 0.0;

        bi9[3][0] = 0.0;
        bi9[3][1] = 0.0;
        bi9[3][2] = 0.0;
        bi9[3][3] = 0.0;
        bi9[3][4] = 0.0;
        bi9[3][5] = 0.0;
        bi9[3][6] = 0.0;
        bi9[3][7] = 0.0;
        bi9[3][8] = 0.0;

        bi9[4][0] = 0.0;
        bi9[4][1] = 0.0;
        bi9[4][2] = 0.0;
        bi9[4][3] = 0.0;
        bi9[4][4] = 0.0;
        bi9[4][5] = 0.0;
        bi9[4][6] = 0.0;
        bi9[4][7] = 0.0;
        bi9[4][8] = 0.0;

        bi9[5][0] = 0.0;
        bi9[5][1] = 0.0;
        bi9[5][2] = 0.0;
        bi9[5][3] = 0.0;
        bi9[5][4] = 0.0;
        bi9[5][5] = 0.0;
        bi9[5][6] = 0.0;
        bi9[5][7] = 0.0;
        bi9[5][8] = 0.0;

        bi9[6][0] = 0.0;
        bi9[6][1] = 0.0;
        bi9[6][2] = 0.0;
        bi9[6][3] = 0.0;
        bi9[6][4] = 0.0;
        bi9[6][5] = 0.0;
        bi9[6][6] = 0.0;
        bi9[6][7] = 0.0;
        bi9[6][8] = 0.0;

        bi9[7][0] = 0.0;
        bi9[7][1] = -0.597_216_760_566_844_7;
        bi9[7][2] = 47.936_048_364_131_67;
        bi9[7][3] = -565.717_041_700_233_3;
        bi9[7][4] = 2_799.622_707_202_858_7;
        bi9[7][5] = -7_032.161_794_919_679;
        bi9[7][6] = 9_404.154_583_498_766;
        bi9[7][7] = -6_374.661_804_344_304;
        bi9[7][8] = 1_720.369_337_666_28;

        bi9[8][0] = 0.0;
        bi9[8][1] = 0.134_984_470_796_006_98;
        bi9[8][2] = -10.834_629_145_944_527;
        bi9[8][3] = 127.864_823_186_993_43;
        bi9[8][4] = -632.777_936_777_218_8;
        bi9[8][5] = 1_589.427_325_412_255_4;
        bi9[8][6] = -2_125.551_246_305_557;
        bi9[8][7] = 1_440.817_483_665_755;
        bi9[8][8] = -388.842_309_780_701_1;

        bi9[9][0] = 0.0;
        bi9[9][1] = 7.290_747_606_753_266e-2;
        bi9[9][2] = -5.851_972_901_033_216;
        bi9[9][3] = 69.062_029_738_762_99;
        bi9[9][4] = -341.774_442_716_209_7;
        bi9[9][5] = 858.477_527_120_137_3;
        bi9[9][6] = -1_148.047_443_579_864_2;
        bi9[9][7] = 778.210_749_640_921_4;
        bi9[9][8] = -210.020_539_601_353_93;

        bi9[10][0] = 0.0;
        bi9[10][1] = 0.128_853_484_734_578_56;
        bi9[10][2] = -10.342_520_980_591_798;
        bi9[10][3] = 122.057_211_066_254_17;
        bi9[10][4] = -604.037_203_213_700_4;
        bi9[10][5] = 1_517.235_637_581_122_8;
        bi9[10][6] = -2_029.008_844_152_904;
        bi9[10][7] = 1_375.375_645_376_321;
        bi9[10][8] = -371.181_116_850_131_56;

        bi9[11][0] = 0.0;
        bi9[11][1] = 0.695_897_172_076_238_7;
        bi9[11][2] = -55.856_705_135_748_31;
        bi9[11][3] = 659.192_634_079_572_8;
        bi9[11][4] = -3_262.215_084_140_846_6;
        bi9[11][5] = 8_194.112_807_588_29;
        bi9[11][6] = -10_958.039_044_672_965;
        bi9[11][7] = 7_427.971_576_643_476;
        bi9[11][8] = -2_004.632_548_946_417_3;

        bi9[12][0] = 0.0;
        bi9[12][1] = 2.617_668_058_132_623e-2;
        bi9[12][2] = -2.101_090_775_094_604_6;
        bi9[12][3] = 24.796_012_566_600_32;
        bi9[12][4] = -122.710_603_910_578_77;
        bi9[12][5] = 308.227_540_243_681_4;
        bi9[12][6] = -412.194_645_100_066_96;
        bi9[12][7] = 279.408_578_064_554_5;
        bi9[12][8] = -75.405_718_003_049_09;

        bi9[13][0] = 0.0;
        bi9[13][1] = 7.845_665_161_262_075e-2;
        bi9[13][2] = -6.297_381_611_696_196;
        bi9[13][3] = 74.318_518_472_037_82;
        bi9[13][4] = -367.787_774_705_65;
        bi9[13][5] = 923.818_459_990_864;
        bi9[13][6] = -1_235.428_287_659_739;
        bi9[13][7] = 837.442_371_605_615_4;
        bi9[13][8] = -226.005_743_111_108_05;

        bi9[14][0] = 0.0;
        bi9[14][1] = 1.743_239_982_411_997e-2;
        bi9[14][2] = -1.399_224_563_421_080_7;
        bi9[14][3] = 16.512_941_882_067_313;
        bi9[14][4] = -81.719_311_330_651_92;
        bi9[14][5] = 205.264_594_249_812_8;
        bi9[14][6] = -274.501_644_179_880_54;
        bi9[14][7] = 186.072_563_019_500_14;
        bi9[14][8] = -50.216_551_375_567_66;

        bi9[15][0] = 0.0;
        bi9[15][1] = 0.0;
        bi9[15][2] = 0.0;
        bi9[15][3] = 0.0;
        bi9[15][4] = 0.0;
        bi9[15][5] = 0.0;
        bi9[15][6] = 0.0;
        bi9[15][7] = 0.0;
        bi9[15][8] = 0.0;

        bi9[16][0] = 0.0;
        bi9[16][1] = -3.120_237_034_739_356_2e-2;
        bi9[16][2] = 2.505_323_513_858_665_6;
        bi9[16][3] = -29.632_099_441_822_69;
        bi9[16][4] = 147.173_574_899_191_83;
        bi9[16][5] = -371.668_262_629_709_6;
        bi9[16][6] = 500.803_253_762_596_74;
        bi9[16][7] = -342.949_046_514_438_8;
        bi9[16][8] = 93.798_458_780_671_25;

        bi9[17][0] = 0.0;
        bi9[17][1] = 1.005_189_184_600_557_6;
        bi9[17][2] = -80.145_247_628_498_57;
        bi9[17][3] = 904.286_287_161_854;
        bi9[17][4] = -4_159.377_900_797_261;
        bi9[17][5] = 9_382.948_440_638_738;
        bi9[17][6] = -10_865.074_362_321_895;
        bi9[17][7] = 6_171.218_731_586_135;
        bi9[17][8] = -1_354.861_137_823_673_3;

        bi9[18][0] = 0.0;
        bi9[18][1] = -1.018_790_304_892_149;
        bi9[18][2] = 81.229_685_434_330_86;
        bi9[18][3] = -916.522_099_840_849_8;
        bi9[18][4] = 4_215.658_051_870_923;
        bi9[18][5] = -9_509.908_233_268_861;
        bi9[18][6] = 11_012.088_661_364_269;
        bi9[18][7] = -6_254.720_911_673_139;
        bi9[18][8] = 1_373.193_636_418_218_8;

        bi9[19][0] = 0.0;
        bi9[19][1] = 1.486_756_383_162_592e-2;
        bi9[19][2] = -1.185_413_256_701_197_5;
        bi9[19][3] = 13.375_128_087_739_489;
        bi9[19][4] = -61.520_574_820_481_606;
        bi9[19][5] = 138.781_422_449_831_18;
        bi9[19][6] = -160.703_267_695_199_86;
        bi9[19][7] = 91.277_333_477_520_88;
        bi9[19][8] = -20.039_495_806_540_547;

        bi9[20][0] = 0.0;
        bi9[20][1] = -2.611_963_106_691_749e-3;
        bi9[20][2] = 0.208_255_752_438_779_78;
        bi9[20][3] = -2.349_769_034_664_475_5;
        bi9[20][4] = 10.808_056_622_683_038;
        bi9[20][5] = -24.381_395_596_370_474;
        bi9[20][6] = 28.232_668_855_390_138;
        bi9[20][7] = -16.035_783_011_964_465;
        bi9[20][8] = 3.520_578_375_594_15;

        bi9[21][0] = 0.0;
        bi9[21][1] = 61.751_365_685_088_68;
        bi9[21][2] = -750.126_891_055_288_4;
        bi9[21][3] = 3_898.685_586_244_842;
        bi9[21][4] = -10_841.822_728_647_458;
        bi9[21][5] = 17_321.575_664_625_165;
        bi9[21][6] = -15_937.704_924_918_136;
        bi9[21][7] = 7_853.047_625_862_507;
        bi9[21][8] = -1_605.405_697_796_720_2;

        bi9[22][0] = 0.0;
        bi9[22][1] = -1.934_298_042_224_096_7;
        bi9[22][2] = 147.588_643_166_560_36;
        bi9[22][3] = -1_181.453_462_187_682;
        bi9[22][4] = 4_124.816_268_317_402;
        bi9[22][5] = -7_708.555_106_592_447;
        bi9[22][6] = 8_052.349_419_898_65;
        bi9[22][7] = -4_443.023_055_054_952;
        bi9[22][8] = 1_010.211_590_494_692_5;

        bi9[23][0] = 0.0;
        bi9[23][1] = 0.912_642_830_786_966_6;
        bi9[23][2] = -71.317_106_090_194_78;
        bi9[23][3] = 695.456_144_607_544_6;
        bi9[23][4] = -2_620.882_193_436_484;
        bi9[23][5] = 4_957.008_141_154_351;
        bi9[23][6] = -5_038.298_085_544_127;
        bi9[23][7] = 2_633.333_935_059_078_6;
        bi9[23][8] = -556.213_478_580_953_7;

        bi9[24][0] = 0.0;
        bi9[24][1] = -0.465_443_997_074_799_17;
        bi9[24][2] = 37.090_998_104_441_54;
        bi9[24][3] = -417.138_222_833_985_5;
        bi9[24][4] = 1_920.758_799_061_188_3;
        bi9[24][5] = -4_409.960_156_149_582;
        bi9[24][6] = 5_383.383_503_749_768;
        bi9[24][7] = -3_353.518_501_772_742;
        bi9[24][8] = 839.849_023_837_986_2;

        bi9[25][0] = 0.0;
        bi9[25][1] = -0.117_645_170_825_521_85;
        bi9[25][2] = 9.481_894_819_354_299;
        bi9[25][3] = -114.915_675_829_831_94;
        bi9[25][4] = 591.319_329_131_246_7;
        bi9[25][5] = -1_559.765_475_086_581_5;
        bi9[25][6] = 2_198.545_841_718_649;
        bi9[25][7] = -1_564.562_048_273_728_4;
        bi9[25][8] = 440.013_778_691_717;

        let c = c.map(|x| T::from_f64(x).unwrap());
        let a = a.map(|row| row.map(|x| T::from_f64(x).unwrap()));
        let b = b.map(|x| T::from_f64(x).unwrap());
        let bh = bh.map(|x| T::from_f64(x).unwrap());
        let bi9 = bi9.map(|row| row.map(|x| T::from_f64(x).unwrap()));

        Self {
            c,
            a,
            b,
            bh: Some(bh),
            bi: Some(bi9),
            er: None,
        }
    }
}