diffurch 0.0.3

//! In this module, [RungeKuttaTable] struct and many known Runge-Kutta methods are defined.

use crate::polynomial;

/// Container for the Butcher's tables and other relevant information about an explicit Runge-Kutta method.
///
/// Generic parameter `S: usize` represents the number of stages of the method, and affects the
/// sizes (hence data layout) of the coefficient tables.
///
/// The values of `a`, which is usually a lower-triangluar 2d-matrix with zero diagonal, are flattened, skipping diagonal and upper-triangle elements.
#[derive(Clone)]
pub struct RungeKuttaTable<const S: usize>
where
    [(); S * (S - 1) / 2]:,
{
    /// Order of the method (to verify and to use in automatic step size control)
    pub order: usize,
    /// Order of the embedded method provided by [RungeKuttaTable::b2], also used in automatic
    /// step size control.
    pub order_embedded: usize,
    /// Order of the dense output scheme
    pub order_interpolant: usize,
    /// a-coefficients of the internal stages, as flat array
    pub a: [f64; S * (S - 1) / 2],
    /// b-coefficients of the final stage
    pub b: [f64; S],
    /// b-coefficients of the final stage of embedded method
    pub b2: [f64; S],
    /// c-coefficients, which are sums of each row of a-coefficients
    pub c: [f64; S],
    /// polynomials that are used in place of b-coefficients, for dense output (interpolation)
    pub bi: [crate::polynomial::Differentiable<fn(f64) -> f64, fn(f64) -> f64>; S],
}

impl<const S: usize> RungeKuttaTable<S>
where
    [(); S * (S - 1) / 2]:,
{
    /// [Self::a] values using two dimensional
    pub fn a_indexed(&self, i: usize, j: usize) -> f64 {
        self.a[i * (i - 1) / 2 + j]
    }
}

/// Euler method (<https://en.wikipedia.org/wiki/Euler_method>), with linear interpolation
pub static EULER: RungeKuttaTable<1> = RungeKuttaTable {
    order: 1,
    order_embedded: 0,
    order_interpolant: 1,
    a: [],
    b: [1.],
    b2: [0.],
    c: [0.],
    bi: [polynomial![0., 1.]],
};

/// Macro declares a static RungeKuttaTable<2> of order 2 with linear interpolantion, and Euler method as an
/// embedded scheme.
/// <https://en.wikipedia.org/wiki/List_of_Runge%E2%80%93Kutta_methods#cite_ref-butcher_1-0>
///
/// # Usage
/// ```
/// #![feature(generic_const_exprs)]
/// use diffurch::{generic_rk_order2, polynomial, rk::RungeKuttaTable};
/// generic_rk_order2!(MIDPOINT, 0.5);
/// generic_rk_order2!(HEUN2, 1.);
/// generic_rk_order2!(RALSTON2, 2. / 3.);
/// ```
#[macro_export]
macro_rules! generic_rk_order2 {
    ($TypeName:ident, $alpha:expr) => {
        /// Order 2 method with linear interpolation and embedded Euler method. See
        /// <https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods>
        pub static $TypeName: RungeKuttaTable<2> = RungeKuttaTable {
            order: 2,
            order_embedded: 1,
            order_interpolant: 1,
            a: [$alpha],
            b: [1. - 0.5 / $alpha, 0.5 / $alpha],
            b2: [1., 0.],
            c: [0., $alpha],
            bi: [
                polynomial![0., 1. - 0.5 / $alpha],
                polynomial![0., 0.5 / $alpha],
            ], // linear interpolation
        };
    };
}

generic_rk_order2!(MIDPOINT, 0.5);
generic_rk_order2!(HEUN2, 1.);
generic_rk_order2!(RALSTON2, 2. / 3.);

/// Macro declares a static RungeKuttaTable<3> of order 3 with linear interpolantion, and embedded order 2 method
/// <https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods>
///
/// # Usage
/// ```
/// #![feature(generic_const_exprs)]
/// use diffurch::{generic_rk_order3, polynomial, rk::RungeKuttaTable};
/// generic_rk_order3!(KUTTA3, 0.5, 1.);
/// generic_rk_order3!(HEUN3, 1. / 3., 2. / 3.);
/// generic_rk_order3!(RALSTON3, 1. / 2., 3. / 4.); // also used in the embedded Bogacki-Shampine
/// generic_rk_order3!(WRAY3, 8. / 15., 2. / 3.);
/// generic_rk_order3!(SSP3, 1., 1. / 2.); // strong stability preserving
/// ```
#[macro_export]
macro_rules! generic_rk_order3 {
    ($TypeName:ident, $alpha:expr, $beta:expr) => {
        /// Order 3 method with linear interpolation and embedded order 2 method. See
        /// <https://en.wikipedia.org/wiki/List_of_Runge–Kutta_methods>.
        /// See in the book "Ernst Hairer , Gerhard Wanner , Syvert P. Nørsett - Solving Ordinary Differential Equations I": Embedded Runge-Kutta Formulas (Methods of order 3(2)).
        pub static $TypeName: RungeKuttaTable<3> = RungeKuttaTable {
            order: 3,
            order_embedded: 2,
            order_interpolant: 1,
            a: [
                $alpha,
                ($beta / $alpha) * ($beta - 3. * $alpha * (1. - $alpha)) / (3. * $alpha - 2.),
                ($beta / $alpha) * ($alpha - $beta) / (3. * $alpha - 2.),
            ],
            b: [
                1. - (3. * $alpha + 3. * $beta - 2.) / (6. * $alpha * $beta),
                (3. * $beta - 2.) / (6. * $alpha * ($beta - $alpha)),
                (2. - 3. * $alpha) / (6. * $beta * ($beta - $alpha)),
            ],
            b2: [1. - 0.5 / $alpha, 0.5 / $alpha, 0.],
            c: [0., $alpha, $beta],
            bi: [
                polynomial![
                    0.,
                    1. - (3. * $alpha + 3. * $beta - 2.) / (6. * $alpha * $beta)
                ],
                polynomial![0., (3. * $beta - 2.) / (6. * $alpha * ($beta - $alpha))],
                polynomial![0., (2. - 3. * $alpha) / (6. * $beta * ($beta - $alpha))],
            ], // linear interpolation
        };
    };
}
generic_rk_order3!(KUTTA3, 0.5, 1.);
generic_rk_order3!(HEUN3, 1. / 3., 2. / 3.);
generic_rk_order3!(RALSTON3, 1. / 2., 3. / 4.); // also used in the embedded Bogacki-Shampine
generic_rk_order3!(WRAY3, 8. / 15., 2. / 3.);
generic_rk_order3!(SSP3, 1., 1. / 2.); // strong stability preserving

/// "The" Runge-Kutta method, with embedded order 2 method, and with order 3 interpolant.
pub static CLASSIC4: RungeKuttaTable<4> = RungeKuttaTable {
    order: 4,
    order_embedded: 2,
    order_interpolant: 3,
    a: [
        0.5, //
        0., 0.5, //
        0.0, 0.0, 1.0,
    ],
    b: [1. / 6., 1. / 3., 1. / 3., 1. / 6.],
    b2: [0., 1., 0., 0.],
    c: [0., 0.5, 0.5, 1.],
    bi: [
        polynomial![0., 1., -1.5, 2. / 3.],
        polynomial![0., 0., 1., -2. / 3.],
        polynomial![0., 0., 1., -2. / 3.],
        polynomial![0., 0., -0.5, 2. / 3.],
    ],
};

/// "The" Runge-Kutta method, with embedded order 3 method, and with order 3 interpolant. Also
/// known as the Zonneveld 4(3) method.
/// See in the book "Ernst Hairer , Gerhard Wanner , Syvert P. Nørsett - Solving Ordinary Differential Equations I": Embedded Runge-Kutta Formulas (Table 4.2. Zonneveld 4(3)).
pub static CLASSIC43: RungeKuttaTable<5> = RungeKuttaTable {
    order: 4,
    order_embedded: 3,
    order_interpolant: 3,
    a: [
        0.5, //
        0.,
        0.5, //
        0.0,
        0.0,
        1.0, //
        5. / 32.,
        7. / 32.,
        13. / 32.,
        -1. / 32., //
    ],
    b: [1. / 6., 1. / 3., 1. / 3., 1. / 6., 0.],
    b2: [-1. / 2., 7. / 3., 7. / 3., 13. / 6., -16. / 3.],
    c: [0., 0.5, 0.5, 1., 0.75],
    bi: [
        polynomial![0., 1., -1.5, 2. / 3.],
        polynomial![0., 0., 1., -2. / 3.],
        polynomial![0., 0., 1., -2. / 3.],
        polynomial![0., 0., -0.5, 2. / 3.],
        polynomial![0.],
    ],
};

/// Dormand-Prince method of order 5, with embedded order 4 method, and with order 4 interpolant.
pub static DP544: RungeKuttaTable<7> = RungeKuttaTable {
    order: 5,
    order_embedded: 4,
    order_interpolant: 4,
    a: [
        0.20000000000000000000000000000000000000000000000000,
        //
        0.07500000000000000000000000000000000000000000000000,
        0.22500000000000000000000000000000000000000000000000,
        //
        0.97777777777777777777777777777777777777777777777778,
        -3.73333333333333333333333333333333333333333333333330,
        3.55555555555555555555555555555555555555555555555560,
        //
        2.95259868922420362749580856576741350403901844231060,
        -11.59579332418838591678097850937357110196616369455900,
        9.82289285169943606157597927145252248132906569120560,
        -0.29080932784636488340192043895747599451303155006859,
        //
        2.84627525252525252525252525252525252525252525252530,
        -10.75757575757575757575757575757575757575757575757600,
        8.90642271774347246045359252906422717743472460453590,
        0.27840909090909090909090909090909090909090909090909,
        -0.27353130360205831903945111492281303602058319039451,
        //
        0.09114583333333333333333333333333333333333333333333,
        0.00000000000000000000000000000000000000000000000000,
        0.44923629829290206648697214734950584007187780772686,
        0.65104166666666666666666666666666666666666666666667,
        -0.32237617924528301886792452830188679245283018867925,
        0.13095238095238095238095238095238095238095238095238,
    ],
    b: [
        0.09114583333333333333333333333333333333333333333333,
        0.00000000000000000000000000000000000000000000000000,
        0.44923629829290206648697214734950584007187780772686,
        0.65104166666666666666666666666666666666666666666667,
        -0.32237617924528301886792452830188679245283018867925,
        0.13095238095238095238095238095238095238095238095238,
        0.,
    ],
    b2: [
        0.08991319444444444444444444444444444444444444444444,
        0.00000000000000000000000000000000000000000000000000,
        0.45348906858340820604971548367774782869122491764001,
        0.61406250000000000000000000000000000000000000000000,
        -0.27151238207547169811320754716981132075471698113208,
        0.08904761904761904761904761904761904761904761904762,
        0.02500000000000000000000000000000000000000000000000,
    ],
    c: [
        0.00000000000000000000000000000000000000000000000000,
        0.20000000000000000000000000000000000000000000000000,
        0.30000000000000000000000000000000000000000000000000,
        0.80000000000000000000000000000000000000000000000000,
        0.88888888888888888888888888888888888888888888888889,
        1.00000000000000000000000000000000000000000000000000,
        1.00000000000000000000000000000000000000000000000000,
    ],
    bi: [
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            1.00000000000000000000000000000000000000000000000000,
            -2.86053866903708863097943370885663105213518085381780,
            3.09957787870912092268605753792811855250234711279230,
            -1.16181058364030928576714728261967728193248499746470,
            0.01391720730161032739385678688152311489865207182347
        ],
        polynomial![0.],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            4.04714149964278557319049904942449527421302289187800,
            -6.34535404693892573505342403022818979479695679525830,
            2.79546508641400508297021164893042296731423395351700,
            -0.04801624082496285462031452077722260665842224240982
        ],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            7.84741007111265892938301126569893156405542561453250,
            -13.50816969460696101391505947117688990840374671710300,
            6.72931750920927857301441847859031845797454992394090,
            -0.41751621904830982181570360644569344695956215470417
        ],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            2.84194470158703464060417964659343605697434957168720,
            -7.54767586295938599838505349903387310042704029578820,
            4.95763672493125298061794541677800406766688093311860,
            -0.57428174280418464170499609263945381666702039769687
        ],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            2.39670292163108553124449083469142506413685252737370,
            -4.74272536284904180356196142354156484858415188009150,
            2.95010386556673177529521224777075926666250808282390,
            -0.47312904339639455059678927796823852983425634915370
        ],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            1.52360049411133393144135467348444915173617479911710,
            -4.32946547433983506636374840484767697183937753380880,
            3.08812946634566833840343278924200648847023067026640,
            -0.28226448611716720348103905787877866836702793557465
        ],
    ],
};

/// Runge-Kutta Tsitouras-Papakostas 6(4)
///
/// SIAM J. Sci. Comput., 20 (1999)  2067-2088
///
/// Coefficients: http://users.uoa.gr/~tsitourasc/rktp64.m
///
/// More from the author: http://users.uoa.gr/~tsitourasc/publications.html
pub static RKTP64: RungeKuttaTable<7> = RungeKuttaTable {
    order: 6,
    order_embedded: 4,
    order_interpolant: 4,
    a: [
        0.14814814814814814814814814814814814814814814814815,
        //
        0.05555555555555555555555555555555555555555555555556,
        0.16666666666666666666666666666666666666666666666667,
        //
        0.19241982507288629737609329446064139941690962099125,
        -0.53134110787172011661807580174927113702623906705539,
        0.76749271137026239067055393586005830903790087463557,
        //
        0.27138264973958333333333333333333333333333333333333,
        -0.28179931640625000000000000000000000000000000000000,
        0.10191932091346153846153846153846153846153846153846,
        0.59599734575320512820512820512820512820512820512821,
        //
        -0.12140681348692272679528027730121494345436084170722,
        0.47761410187445690404270285927090660818471469359043,
        0.12192296968479080920271208457739825271063725338342,
        0.00820786686248269381285345905535723094994297948894,
        0.28289264429596155050624264362832208237829668447521,
        //
        0.32310946589106292966684294024325753569539925965098,
        -0.61039132734003172924378635642517186673717609730301,
        0.45846867541639319976612888047255080412929454343221,
        0.57505740806711566278133278660922926335638856572569,
        -0.57379234522267681781745625845989491691225880929347,
        0.82754812318813675484693800756002918046835253778760,
    ],
    b: [
        0.07277777777777777777777777777777777777777777777778,
        0.00000000000000000000000000000000000000000000000000,
        0.28752127070690503526324421846809906511399048712482,
        0.18974846220396832187710941882243328294496258901153,
        0.10581736348682550735167973519824811036097403449285,
        0.26909544328484081804764916719375922411975542905334,
        0.07503968253968253968253968253968253968253968253968,
    ],
    b2: [
        0.10322666047518118524035683798997408464864086165861,
        0.00000000000000000000000000000000000000000000000000,
        0.15611542056134071025476537742246069068941506933613,
        0.38634918851063907802720917783777796648011517818308,
        -0.12073095208684351320487107578989528150071079171750,
        0.40000000000000000000000000000000000000000000000000,
        0.07503968253968253968253968253968253968253968253968,
    ],
    c: [
        0.00000000000000000000000000000000000000000000000000,
        0.14814814814814814814814814814814814814814814814815,
        0.22222222222222222222222222222222222222222222222222,
        0.42857142857142857142857142857142857142857142857143,
        0.68750000000000000000000000000000000000000000000000,
        0.76923076923076923076923076923076923076923076923077,
        1.00000000000000000000000000000000000000000000000000,
    ],
    bi: [
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            1.00000000000000000000000000000000000000000000000000,
            -3.18888888888888888888888888888888888888888888888890,
            3.62962962962962962962962962962962962962962962962960,
            -1.36796296296296296296296296296296296296296296296300
        ],
        polynomial![0.],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            3.13664097096932917828440216499917992455305888141710,
            -4.87567656224372642283090044284074134820403477119900,
            2.02655686198130227980974249630966048876496637690670
        ],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            1.44306660772178013557323902151488358384910109048040,
            -2.68732193732193732193732193732193732193732193732190,
            1.43400379180412550824119233462948702103318343585310
        ],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            -1.84105678504031566306399039286326985760850917824670,
            5.43416252072968490878938640132669983416252072968490,
            -3.48728837220254373837371627326518186619303751694540
        ],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            0.26909544328484081804764916719375922411975542905334
        ],
        polynomial![
            0.00000000000000000000000000000000000000000000000000,
            0.00000000000000000000000000000000000000000000000000,
            0.45023809523809523809523809523809523809523809523810,
            -1.50079365079365079365079365079365079365079365079370,
            1.12559523809523809523809523809523809523809523809520
        ],
    ],
};

/// High order interpolant scheme by Charalampos Tsitouras
///
/// See the paper: https://www.researchgate.net/publication/220393895_Runge-Kutta_interpolants_for_high_precision_computations
///
/// Mathematica file with coefficients: http://users.uoa.gr/~tsitourasc/NumericalAlgorithms.txt
///
/// More from the author: http://users.uoa.gr/~tsitourasc/publications.html
pub static RK98: RungeKuttaTable<26> = RungeKuttaTable {
    order: 9,
    order_embedded: 8,
    order_interpolant: 9,

    a: [
        0.02173913043478260869565217391304347826086956521739,
        //
        -0.11698050118114486205818241524969622410384223979750,
        0.21327631165914552875931243204789548369469882662606,
        //
        0.03611092892925025001292375629932472234657122006071,
        0.00000000000000000000000000000000000000000000000000,
        0.10833278678775075003877126889797416703971366018213,
        //
        1.57329743908138605107331820072051125641643664351840,
        0.00000000000000000000000000000000000000000000000000,
        -5.98400943754042002888532938159655553518127689802840,
        4.93277082198844574251789353381722074935307554862770,
        //
        0.05052046351120380909008334360006234635511181026904,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.17686653884807108146683657390397612041041458603582,
        0.00103743376935980522339467349390418494016089864590,
        //
        0.10543148021953768958529340893598138847228076756097,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        -0.16042415162569842979496486916719383758722635620698,
        0.11643956912829316045688724281285250393566139141568,
        0.48215663817720491194449759844838932144562487430754,
        //
        0.07148407148407148407148407148407148407148407148407,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.32971116090443908023196389566296464803047147789829,
        0.24216141096813279233990867620960722454140109397428,
        //
        0.07162368881118881118881118881118881118881118881119,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.32859867301674234161492268975519694162614616072449,
        0.11622213117906185418927311444060725417804964347131,
        -0.03392701048951048951048951048951048951048951048951,
        //
        0.04861540768024729180628870095388582672433144665591,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.03998502200331629058445317782406268415701330226122,
        0.10715724786209388876739304914053506956742307410965,
        -0.02177735985419485163815426357369818120744539675290,
        -0.10579849950964443770179884616296721742314060809206,
        //
        -0.02540141041535143673515871979014924329181062727138,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.03333333333333333333333333333333333333333333333333,
        -0.16404854760069182073503553020238782592593403647268,
        0.03410548898794737788891414566528526831566085030511,
        0.15836825014108792658008718465091487655106285177706,
        0.21425115805975734472868683695127609710041268915582,
        //
        0.00584833331460742801095934302256470549308679646338,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        0.00000000000000000000000000000000000000000000000000,
        -0.53954170547283522916525526480339109942148038774928,
        0.20128430845560909506500331018201158854604633245476,
        0.04347222773254789483240207937678906296377158021513,
        -0.00402998571475307250775349983910179275996883814082,
        0.16541535721570612771420482097898952379873117545074,
        0.79491862412512344573322086551518180067011337498503,
        //
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#[cfg(test)]
mod tests {
    use super::*;

    fn interpolation_continuity_error<const S: usize>(rk: &RungeKuttaTable<S>) -> f64
    where
        [(); S * (S - 1) / 2]:,
    {
        let mut max = 0f64;
        for i in 0..S {
            max = max.max((rk.b[i] - rk.bi[i](1.)).abs());
            max = max.max(rk.bi[i](0.).abs());
        }
        max
    }

    fn c_is_sum_of_a_error<const S: usize>(rk: &RungeKuttaTable<S>) -> f64
    where
        [(); S * (S - 1) / 2]:,
    {
        let mut max = 0f64;
        for i in 0..S {
            let sum = ((i * i - i) / 2..).take(i).map(|j| rk.a[j]).sum::<f64>();
            let diff = (rk.c[i] - sum).abs();
            max = f64::max(max, diff);
        }
        max
    }

    fn order_conditions_error<const ORDER: usize, const S: usize>(
        rk: &RungeKuttaTable<S>,
        _: [(); ORDER],
    ) -> f64
    where
        [(); S * (S - 1) / 2]:,
    {
        // On derivation and formulas for order conditions, see
        // E. Hairer, G. Wanner, and S. P. Nørsett, Solving ordinary differential equations I, vol. 8. in Springer Series in Computational Mathematics, vol. 8. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993. doi: 10.1007/978-3-540-78862-1.
        let mut max_error = 0.;

        // order is q in the mentioned text
        for order in 1..=ORDER {
            let advance_tree = |tree: &mut [usize]| -> bool {
                let mut i_of_increased = order - 1;

                // while cannot increase in this index
                while i_of_increased > 0 && tree[i_of_increased] + 1 >= i_of_increased {
                    i_of_increased -= 1;
                }

                if i_of_increased > 0 {
                    tree[i_of_increased] += 1;
                    for i in (i_of_increased + 1)..order {
                        tree[i] = tree[i_of_increased];
                    }
                    return true;
                } else {
                    return false;
                }
            };

            let fix_indexes = |tree: &[usize], indexes: &mut [usize]| {
                for i in (1..order).rev() {
                    indexes[tree[i]] = usize::max(indexes[tree[i]], indexes[i] + 1);
                }
            };

            let advance_indexes = |tree: &[usize], indexes: &mut [usize]| -> bool {
                let mut i_of_increased = order - 1;
                while i_of_increased > 0
                    && indexes[i_of_increased] + 1 >= indexes[tree[i_of_increased]]
                {
                    i_of_increased -= 1;
                }

                indexes[i_of_increased] += 1;
                for i in (i_of_increased + 1)..order {
                    indexes[i] = 0;
                }
                for i in ((i_of_increased + 1)..order).rev() {
                    indexes[tree[i]] = usize::max(indexes[tree[i]], indexes[i] + 1);
                }
                if i_of_increased > 0 {
                    return true;
                } else {
                    return false;
                }
            };

            let gamma = |tree: &[usize]| -> f64 {
                let mut rho = vec![0f64; order];
                for i in (1..order).rev() {
                    rho[i] += 1.;
                    rho[tree[i]] += rho[i];
                }
                rho[0] += 1.;

                rho.iter().product()
            };

            let mut tree = [0usize; ORDER];

            // index[0] in 0..order, index[j] in 0..index[tree[j]]
            let mut indexes = [0usize; ORDER];

            loop
            /* for all trees */
            {
                fix_indexes(&tree, &mut indexes);

                let mut b_phi_sum = 0.;
                let mut b_phi_sum_embedded = 0.;

                println!("{:?}", &tree[..order]);
                for j in indexes[0]..S {
                    // j = 0 => sums are empty
                    indexes.fill(0);
                    indexes[0] = j;
                    fix_indexes(&tree, &mut indexes);

                    let mut phi_j = 0.;

                    loop
                    /* for all index packs starting with j */
                    {
                        // println!("\t{:?}", indexes);
                        phi_j += (1..order).fold(1., |acc, i| {
                            acc * rk.a_indexed(indexes[tree[i]], indexes[i])
                        });

                        if !advance_indexes(&tree, &mut indexes) {
                            indexes.fill(0);
                            break;
                        }
                    }
                    b_phi_sum += phi_j * rk.b[j];
                    b_phi_sum_embedded += phi_j * rk.b2[j];
                }

                let error = f64::abs(b_phi_sum - 1. / gamma(&tree));
                let error_embedded = f64::abs(b_phi_sum_embedded - 1. / gamma(&tree));
                max_error = f64::max(max_error, error);
                if order <= rk.order_embedded {
                    max_error = f64::max(max_error, error_embedded);
                }

                println!("\t{:?}", error);
                println!("\t{:?}", error_embedded);

                if !advance_tree(&mut tree) {
                    break;
                }
            }
        }

        max_error
    }

    macro_rules! test_rk {
        ($name:ident, $RK:path, $tolerance:expr) => {
            #[test]
            fn $name() {
                // assert_a_has_correct_sizes(&$RK);
                assert!(interpolation_continuity_error(&$RK) < $tolerance);
                assert!(c_is_sum_of_a_error(&$RK) < $tolerance);
                assert!(order_conditions_error(&$RK, [(); $RK.order]) < $tolerance);
            }
        };
    }

    test_rk!(rk1_euler, EULER, 1e-15);

    test_rk!(rk2_heun, HEUN2, 1e-15);
    test_rk!(rk2_ralston, RALSTON2, 1e-15);
    test_rk!(rk2_midpoint, MIDPOINT, 1e-15);

    test_rk!(rk3_ssp, SSP3, 1e-15);
    test_rk!(rk3_heun, HEUN3, 1e-15);
    test_rk!(rk3_wray, WRAY3, 1e-15);
    test_rk!(rk3_kutta, KUTTA3, 1e-15);
    test_rk!(rk3_ralston, RALSTON3, 1e-15);

    test_rk!(rk4_classic, CLASSIC4, 1e-15);
    test_rk!(rk4_classic_dense, CLASSIC43, 1e-15);

    test_rk!(dormand_prince, DP544, 1e-11);

    test_rk!(rktp64, RKTP64, 1e-15);

    #[test]
    fn rk98_exept_order_conditions() {
        // assert_a_has_correct_sizes(&RK98);
        assert!(interpolation_continuity_error(&RK98) < 1e-9);
        assert!(c_is_sum_of_a_error(&RK98) < 1e-9);
        assert!(order_conditions_error(&RK98, [(); 5]) < 1e-7);
    }

    #[test]
    #[ignore] // because takes too long (1.5 hours for 7th order or smth)
    fn rk98_order_conditions() {
        assert!(order_conditions_error(&RK98, [(); RK98.order]) < 1e-15);
    }
}