// TODO A future improvement can be to use the Gauss–Kronrod rules
// to estimate the error and thus choose a number of constants based
// on the error. Maybe not.

// TODO In future, the constants can be calculated and cached so we can
// choose any value for the order.

// TODO - to limit rounding error do pairwise addition of terms

// TODO order abscissas

// TODO - auto calc abscissas and weights (on first call to function only)


/** 
 * Numerically integrates the given function using the Gaussian Quadrature
 * method.
 * 
 * See https://en.wikipedia.org/wiki/Gaussian_quadrature
 * See http://pomax.github.io/bezierinfo/#arclength
 * @param f The univariate function to be integrated
 * @param interval The integration interval
 * @param order Can be 2, 4, 8, or 16. Higher values give more accurate results 
 * but is slower - defaults to 16.
 */
function gaussQuadrature(
        f: (x: number) => number, 
        interval: number[], 
        order: 2|4|8|16|64 = 16): number {

	if (interval[0] === interval[1]) { return 0; }
	
	const { weights, abscissas } = GAUSS_CONSTANTS[order];
	const [a, b] = interval;

	let result = 0;
	const m1 = (b - a) / 2;
	const m2 = (b + a) / 2;
	for (let i=0; i<=order-1; i++) {
		result += weights[i] * f(m1*abscissas[i] + m2);
	}	
	
	return m1 * result;
}


// The Gaussian Legendre Quadrature method constants. 
const GAUSS_CONSTANTS: { [index:number]: { abscissas: number[]; weights: number[]; } } = {
	2: { 
		weights: [1, 1], 
		abscissas: [-0.5773502691896257, 0.5773502691896257] 
	},
	4: { 
		weights: [
			0.6521451548625461, 0.6521451548625461, 
			0.3478548451374538, 0.3478548451374538
		],
		abscissas: [
			-0.3399810435848563, 0.3399810435848563, 
			-0.8611363115940526, 0.8611363115940526
		]
	},
	8: {
		weights: [
			0.3626837833783620, 0.3626837833783620, 
			0.3137066458778873, 0.3137066458778873,
		    0.2223810344533745, 0.2223810344533745, 
		    0.1012285362903763, 0.1012285362903763],
		abscissas: [
			-0.1834346424956498, 0.1834346424956498, 
			-0.5255324099163290, 0.5255324099163290,
		    -0.7966664774136267, 0.7966664774136267, 
		    -0.9602898564975363, 0.9602898564975363
		]
	},
	// Taken from http://keisan.casio.com/exec/system/1330940731
	16: {
		weights: [
			0.0271524594117540948518,
			0.062253523938647892863,
			0.0951585116824927848099,
			0.1246289712555338720525,
			0.1495959888165767320815,
			0.169156519395002538189,
			0.182603415044923588867,
			0.189450610455068496285,
			0.1894506104550684962854,
			0.182603415044923588867,
			0.1691565193950025381893,
			0.149595988816576732081,
			0.124628971255533872053,
			0.095158511682492784809,
			0.062253523938647892863,
			0.027152459411754094852
		],
		abscissas: [
			-0.989400934991649932596,
			-0.944575023073232576078,
			-0.86563120238783174388,
			-0.7554044083550030338951,
			-0.6178762444026437484467,
			-0.4580167776572273863424,
			-0.28160355077925891323,
			-0.0950125098376374401853,
			0.0950125098376374401853,
			0.28160355077925891323,
		    0.4580167776572273863424,
			0.617876244402643748447,
			0.755404408355003033895,
			0.8656312023878317438805,
			0.944575023073232576078,
			0.989400934991649932596
		],
	},
	64: {
		weights: [
			0.048690957009139724,
			0.048690957009139724,
			0.04857546744150343,
			0.04857546744150343,
			0.048344762234802954,
			0.048344762234802954,
			0.04799938859645831,
			0.04799938859645831,
			0.04754016571483031,
			0.04754016571483031,
			0.04696818281621002,
			0.04696818281621002,
			0.046284796581314416,
			0.046284796581314416,
			0.04549162792741814,
			0.04549162792741814,
			0.044590558163756566,
			0.044590558163756566,
			0.04358372452932345,
			0.04358372452932345,
			0.04247351512365359,
			0.04247351512365359,
			0.04126256324262353,
			0.04126256324262353,
			0.03995374113272034,
			0.03995374113272034,
			0.038550153178615626,
			0.038550153178615626,
			0.03705512854024005,
			0.03705512854024005,
			0.035472213256882386,
			0.035472213256882386,
			0.033805161837141606,
			0.033805161837141606,
			0.03205792835485155,
			0.03205792835485155,
			0.030234657072402478,
			0.030234657072402478,
			0.028339672614259483,
			0.028339672614259483,
			0.02637746971505466,
			0.02637746971505466,
			0.024352702568710874,
			0.024352702568710874,
			0.022270173808383253,
			0.022270173808383253,
			0.02013482315353021,
			0.02013482315353021,
			0.017951715775697343,
			0.017951715775697343,
			0.015726030476024718,
			0.015726030476024718,
			0.013463047896718643,
			0.013463047896718643,
			0.011168139460131128,
			0.011168139460131128,
			0.008846759826363947,
			0.008846759826363947,
			0.006504457968978363,
			0.006504457968978363,
			0.004147033260562468,
			0.004147033260562468,
			0.001783280721696433,
			0.001783280721696433
		],
		abscissas: [
			-0.024350292663424433,
			0.024350292663424433,
			-0.07299312178779904,
			0.07299312178779904,
			-0.12146281929612056,
			0.12146281929612056,
			-0.16964442042399283,
			0.16964442042399283,
			-0.21742364374000708,
			0.21742364374000708,
			-0.2646871622087674,
			0.2646871622087674,
			-0.31132287199021097,
			0.31132287199021097,
			-0.3572201583376681,
			0.3572201583376681,
			-0.4022701579639916,
			0.4022701579639916,
			-0.4463660172534641,
			0.4463660172534641,
			-0.48940314570705296,
			0.48940314570705296,
			-0.5312794640198946,
			0.5312794640198946,
			-0.571895646202634,
			0.571895646202634,
			-0.6111553551723933,
			0.6111553551723933,
			-0.6489654712546573,
			0.6489654712546573,
			-0.6852363130542333,
			0.6852363130542333,
			-0.7198818501716109,
			0.7198818501716109,
			-0.7528199072605319,
			0.7528199072605319,
			-0.7839723589433414,
			0.7839723589433414,
			-0.8132653151227975,
			0.8132653151227975,
			-0.8406292962525803, 
			0.8406292962525803, 
			-0.8659993981540928, 
			0.8659993981540928, 
			-0.8893154459951141, 
			0.8893154459951141, 
			-0.9105221370785028, 
			0.9105221370785028, 
			-0.9295691721319396, 
			0.9295691721319396, 
			-0.9464113748584028, 
			0.9464113748584028,
			-0.9610087996520538,
			0.9610087996520538,
			-0.973326827789911,
			0.973326827789911,
			-0.983336253884626,
			0.983336253884626,
			-0.9910133714767443,
			0.9910133714767443,
			-0.9963401167719553,
			0.9963401167719553,
			-0.9993050417357722, 
			0.9993050417357722
		]
	}
};


export { gaussQuadrature }