/*
    PIP - Platform Independent Primitives
    Math
    Copyright (C) 2011  Ivan Pelipenko peri4ko@gmail.com

    This program is free software: you can redistribute it and/or modify
    it under the terms of the GNU General Public License as published by
    the Free Software Foundation, either version 3 of the License, or
    (at your option) any later version.

    This program is distributed in the hope that it will be useful,
    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
    GNU General Public License for more details.

    You should have received a copy of the GNU General Public License
    along with this program.  If not, see <http://www.gnu.org/licenses/>.
*/

#ifndef PIMATH_H
#define PIMATH_H

#include "picontainers.h"
#ifndef QNX
# include <cmath>
# include <complex>
#else
# include <math.h>
# include <complex.h>
#endif
#ifdef CC_VC
#define M_PI 3.14159265358979323846
#endif

#define M_2PI 6.28318530717958647692
#define M_PI_3 1.04719755119659774615

using std::complex;

typedef complex<int> complexi;
typedef complex<float> complexf;
typedef complex<double> complexd;
typedef complex<ldouble> complexld;
const complexld complexld_i(0., 1.);
const complexld complexld_0(0.);
const complexld complexld_1(1.);
const complexd complexd_i(0., 1.);
const complexd complexd_0(0.);
const complexd complexd_1(1.);

const double deg2rad = atan(1.) / 45.;
const double rad2deg = 45. / atan(1.);

inline int pow2(const int p) {return 1 << p;}
inline double sqr(const double & v) {return v * v;}
inline double sinc(const double & v) {double t = M_PI * v; return sin(t) / t;}
inline complexd round(const complexd & c) {return complexd(round(c.real()), round(c.imag()));}
inline complexd floor(const complexd & c) {return complexd(floor(c.real()), floor(c.imag()));}
inline complexd ceil(const complexd & c) {return complexd(ceil(c.real()), ceil(c.imag()));}
inline complexd atanc(const complexd & c) {return -complexd(-0.5, 1.) * log((complexd_1 + complexd_i * c) / (complexd_1 - complexd_i * c));}
inline complexd asinc(const complexd & c) {return -complexd_i * log(complexd_i * c + sqrt(complexd_1 - c * c));}
inline complexd acosc(const complexd & c) {return -complexd_i * log(c + complexd_i * sqrt(complexd_1 - c * c));}
#ifdef QNX
inline complexd tan(const complexd & c) {return sin(c) / cos(c);}
inline complexd tanh(const complexd & c) {return sinh(c) / cosh(c);}
inline complexd log2(const complexd & c) {return log(c) / M_LN2;}
inline complexd log10(const complexd & c) {return log(c) / M_LN10;}
inline double j0(const double & v) {v;}
inline double j1(const double & v) {v;}
inline double jn(const int & n, const double & v) {v;}
inline double y0(const double & v) {v;}
inline double y1(const double & v) {v;}
inline double yn(const int & n, const double & v) {v;}
#endif

template<uint Cols, uint Rows, typename Type>
class PIMathMatrixT;

/// Vector templated

#define PIMV_FOR(v, s) for (uint v = s; v < Size; ++v)

template<uint Size, typename Type = double>
class PIMathVectorT {
	typedef PIMathVectorT<Size, Type> _CVector;
public:
	PIMathVectorT() {resize(Size);}
	//PIMathVectorT(Type val) {resize(Size); PIMV_FOR(i, 0) c[i] = val;}
	PIMathVectorT(Type fval, ...) {resize(Size); c[0] = fval; va_list vl; va_start(vl, fval); PIMV_FOR(i, 1) c[i] = va_arg(vl, Type); va_end(vl);}
	PIMathVectorT(const PIVector<Type> & val) {resize(Size); PIMV_FOR(i, 0) c[i] = val[i];}
	PIMathVectorT(const _CVector & st, const _CVector & fn) {resize(Size); set(st, fn);}

	uint size() const {return Size;}
	_CVector & fill(const Type & v) {PIMV_FOR(i, 0) c[i] = v; return *this;}
	_CVector & set(Type fval, ...) {c[0] = fval; va_list vl; va_start(vl, fval); PIMV_FOR(i, 1) c[i] = va_arg(vl, Type); va_end(vl); return *this;}
	_CVector & set(const _CVector & st, const _CVector & fn) {PIMV_FOR(i, 0) c[i] = fn[i] - st[i]; return *this;}
	_CVector & move(const Type & v) {PIMV_FOR(i, 0) c[i] += v; return *this;}
	_CVector & move(const _CVector & v) {PIMV_FOR(i, 0) c[i] += v[i]; return *this;}
	_CVector & move(Type fval, ...) {c[0] += fval; va_list vl; va_start(vl, fval); PIMV_FOR(i, 1) c[i] += va_arg(vl, Type); va_end(vl); return *this;}
	Type lengthSqr() const {Type tv(0); PIMV_FOR(i, 0) tv += (c[i] * c[i]); return tv;}
	Type length() const {return sqrt(lengthSqr());}
	Type manhattanLength() const {Type tv(0); PIMV_FOR(i, 0) tv += fabs(c[i]); return tv;}
	Type angleCos(const _CVector & v) const {Type tv = v.length() * length(); return (tv == Type(0) ? Type(0) : ((*this) ^ v) / tv);}
	Type angleSin(const _CVector & v) const {Type tv = angleCos(v); return sqrt(Type(1) - tv * tv);}
	Type angleRad(const _CVector & v) const {return acos(angleCos(v));}
	Type angleDeg(const _CVector & v) const {return acos(angleCos(v)) * rad2deg;}
	_CVector projection(const _CVector & v) {Type tv = v.length(); return (tv == Type(0) ? _CVector() : v * (((*this) ^ v) / tv));}
	_CVector & normalize() {Type tv = length(); if (tv == Type(1)) return *this; PIMV_FOR(i, 0) c[i] /= tv; return *this;}
	_CVector normalized() {_CVector tv(*this); tv.normalize(); return tv;}
	bool isNull() const {PIMV_FOR(i, 0) if (c[i] != Type(0)) return false; return true;}
	bool isOrtho(const _CVector & v) const {return ((*this) ^ v) == Type(0);}

	Type & at(uint index) {return c[index];}
	Type at(uint index) const {return c[index];}
	Type & operator [](uint index) {return c[index];}
	Type operator [](uint index) const {return c[index];}
	void operator =(const _CVector & v) {c = v.c;}
	bool operator ==(const _CVector & v) const {PIMV_FOR(i, 0) if (c[i] != v[i]) return false; return true;}
	bool operator !=(const _CVector & v) const {return !(*this == c);}
	void operator +=(const _CVector & v) {PIMV_FOR(i, 0) c[i] += v[i];}
	void operator -=(const _CVector & v) {PIMV_FOR(i, 0) c[i] -= v[i];}
	void operator *=(const Type & v) {PIMV_FOR(i, 0) c[i] *= v;}
	void operator *=(const _CVector & v) {PIMV_FOR(i, 0) c[i] *= v[i];}
	void operator /=(const Type & v) {PIMV_FOR(i, 0) c[i] /= v;}
	void operator /=(const _CVector & v) {PIMV_FOR(i, 0) c[i] /= v[i];}
	_CVector operator -() const {_CVector tv; PIMV_FOR(i, 0) tv[i] = -c[i]; return tv;}
	_CVector operator +(const _CVector & v) const {_CVector tv = _CVector(*this); PIMV_FOR(i, 0) tv[i] += v[i]; return tv;}
	_CVector operator -(const _CVector & v) const {_CVector tv = _CVector(*this); PIMV_FOR(i, 0) tv[i] -= v[i]; return tv;}
	_CVector operator *(const Type & v) const {_CVector tv = _CVector(*this); PIMV_FOR(i, 0) tv[i] *= v; return tv;}
	_CVector operator /(const Type & v) const {_CVector tv = _CVector(*this); PIMV_FOR(i, 0) tv[i] /= v; return tv;}
	_CVector operator *(const _CVector & v) const {if (Size > 3) return _CVector(); _CVector tv; tv.fill(Type(1)); tv[0] = c[1]*v[2] - v[1]*c[2]; tv[1] = v[0]*c[2] - c[0]*v[2]; tv[2] = c[0]*v[1] - v[0]*c[1]; return tv;}
	Type operator ^(const _CVector & v) const {Type tv(0); PIMV_FOR(i, 0) tv += c[i] * v[i]; return tv;}

	operator PIMathMatrixT<1, Size, Type>() {return PIMathMatrixT<1, Size, Type>(c);}
	Type distToLine(const _CVector & lp0, const _CVector & lp1) {
		_CVector a(lp0, lp1), b(lp0, *this), c(lp1, *this);
		Type f = fabs(a[0]*b[1] - a[1]*b[0]) / a.length();//, s = b.length() + c.length() - a.length();
		return f;}

	template<uint Size1, typename Type1> /// vector {Size, Type} to vector {Size1, Type1}
	PIMathVectorT<Size1, Type1> turnTo() {PIMathVectorT<Size1, Type1> tv; uint sz = piMin<uint>(Size, Size1); for (uint i = 0; i < sz; ++i) tv[i] = c[i]; return tv;}

private:
	void resize(uint size, const Type & new_value = Type()) {c.resize(size, new_value);}
	PIVector<Type> c;

};

template<uint Size, typename Type>
inline std::ostream & operator <<(std::ostream & s, const PIMathVectorT<Size, Type> & v) {s << '{'; PIMV_FOR(i, 0) {s << v[i]; if (i < Size - 1) s << ", ";} s << '}'; return s;}
template<uint Size, typename Type>
inline bool operator ||(const PIMathVectorT<Size, Type> & f, const PIMathVectorT<Size, Type> & s) {return (f * s).isNull();}

//template<uint Size0, typename Type0 = double, uint Size1 = Size0, typename Type1 = Type0> /// vector {Size0, Type0} to vector {Size1, Type1}
//inline operator PIMathVectorT<Size1, Type1>(const PIMathVectorT<Size0, Type0> & v) {PIMathVectorT<Size1, Type1> tv; uint sz = piMin<uint>(Size0, Size1); for (uint i = 0; i < sz; ++i) tv[i] = v[i]; return tv;}

typedef PIMathVectorT<2u, int> PIMathVectorT2i;
typedef PIMathVectorT<3u, int> PIMathVectorT3i;
typedef PIMathVectorT<4u, int> PIMathVectorT4i;
typedef PIMathVectorT<2u, double> PIMathVectorT2d;
typedef PIMathVectorT<3u, double> PIMathVectorT3d;
typedef PIMathVectorT<4u, double> PIMathVectorT4d;

/// Matrix templated

#define PIMM_FOR(c, r) for (uint c = 0; c < Cols; ++c) { for (uint r = 0; r < Rows; ++r) {
#define PIMM_FOR_WB(c, r) for (uint c = 0; c < Cols; ++c) for (uint r = 0; r < Rows; ++r) // without brakes
#define PIMM_FOR_I(c, r) for (uint r = 0; r < Rows; ++r) { for (uint c = 0; c < Cols; ++c) {
#define PIMM_FOR_I_WB(c, r) for (uint r = 0; r < Rows; ++r) for (uint c = 0; c < Cols; ++c) // without brakes
#define PIMM_FOR_C(v) for (uint v = 0; v < Cols; ++v)
#define PIMM_FOR_R(v) for (uint v = 0; v < Rows; ++v)

template<uint Cols, uint Rows = Cols, typename Type = double>
class PIMathMatrixT {
	typedef PIMathMatrixT<Cols, Rows, Type> _CMatrix;
	typedef PIMathMatrixT<Rows, Cols, Type> _CMatrixI;
	typedef PIMathVectorT<Rows, Type> _CMCol;
	typedef PIMathVectorT<Cols, Type> _CMRow;
public:
	PIMathMatrixT() {resize(Cols, Rows);}
	PIMathMatrixT(Type fval, ...) {resize(Cols, Rows); va_list vl; va_start(vl, fval); PIMM_FOR_I_WB(c, r) m[c][r] = (r + c == 0 ? fval : va_arg(vl, Type)); va_end(vl);}
	PIMathMatrixT(const PIVector<Type> & val) {resize(Cols, Rows); int i = 0; PIMM_FOR_I_WB(c, r) m[c][r] = val[i++];}

	static _CMatrix identity() {_CMatrix tm = _CMatrix(); PIMM_FOR_WB(c, r) tm.m[c][r] = (c == r ? Type(1) : Type(0)); return tm;}

	uint cols() const {return Cols;}
	uint rows() const {return Rows;}
	_CMCol col(uint index) {_CMCol tv; PIMM_FOR_R(i) tv[i] = m[index][i]; return tv;}
	_CMRow row(uint index) {_CMRow tv; PIMM_FOR_C(i) tv[i] = m[i][index]; return tv;}
	_CMatrix & setCol(uint index, const _CMCol & v) {PIMM_FOR_R(i) m[index][i] = v[i]; return *this;}
	_CMatrix & setRow(uint index, const _CMRow & v) {PIMM_FOR_C(i) m[i][index] = v[i]; return *this;}
	_CMatrix & swapRows(uint r0, uint r1) {Type t; PIMM_FOR_C(i) {t = m[i][r0]; m[i][r0] = m[i][r1]; m[i][r1] = t;} return *this;}
	_CMatrix & swapCols(uint c0, uint c1) {Type t; PIMM_FOR_R(i) {t = m[c0][i]; m[c0][i] = m[c1][i]; m[c1][i] = t;} return *this;}
	_CMatrix & fill(const Type & v) {PIMM_FOR_WB(c, r) m[c][r] = v; return *this;}
	//inline _CMatrix & set(Type fval, ...) {m[0] = fval; va_list vl; va_start(vl, fval); PIMV_FOR(i, 1) m[i] = va_arg(vl, Type); va_end(vl); return *this;}
	//inline void normalize() {Type tv = length(); if (tv == Type(1)) return; PIMV_FOR(i, 0) m[i] /= tv;}
	bool isSquare() const {return cols() == rows();}
	bool isIdentity() const {PIMM_FOR_WB(c, r) if ((c == r) ? m[c][r] != Type(1) : m[c][r] != Type(0)) return false; return true;}
	bool isNull() const {PIMM_FOR_WB(c, r) if (m[c][r] != Type(0)) return false; return true;}

	Type & at(uint col, uint row) {return m[col][row];}
	Type at(uint col, uint row) const {return m[col][row];}
	PIVector<Type> & operator [](uint col) {return m[col];}
	PIVector<Type> operator [](uint col) const {return m[col];}
	void operator =(const _CMatrix & sm) {m = sm.m;}
	bool operator ==(const _CMatrix & sm) const {PIMM_FOR_WB(c, r) if (m[c][r] != sm.m[c][r]) return false; return true;}
	bool operator !=(const _CMatrix & sm) const {return !(*this == sm);}
	void operator +=(const _CMatrix & sm) {PIMM_FOR_WB(c, r) m[c][r] += sm.m[c][r];}
	void operator -=(const _CMatrix & sm) {PIMM_FOR_WB(c, r) m[c][r] -= sm.m[c][r];}
	void operator *=(const Type & v) {PIMM_FOR_WB(c, r) m[c][r] *= v;}
	void operator /=(const Type & v) {PIMM_FOR_WB(c, r) m[c][r] /= v;}
	_CMatrix operator -() {_CMatrix tm; PIMM_FOR_WB(c, r) tm.m[c][r] = -m[c][r]; return tm;}
	_CMatrix operator +(const _CMatrix & sm) {_CMatrix tm = _CMatrix(*this); PIMM_FOR_WB(c, r) tm.m[c][r] += sm.m[c][r]; return tm;}
	_CMatrix operator -(const _CMatrix & sm) {_CMatrix tm = _CMatrix(*this); PIMM_FOR_WB(c, r) tm.m[c][r] -= sm.m[c][r]; return tm;}
	_CMatrix operator *(const Type & v) {_CMatrix tm = _CMatrix(*this); PIMM_FOR_WB(c, r) tm.m[c][r] *= v; return tm;}
	_CMatrix operator /(const Type & v) {_CMatrix tm = _CMatrix(*this); PIMM_FOR_WB(c, r) tm.m[c][r] /= v; return tm;}

	_CMatrix & toUpperTriangular(bool * ok = 0) {
		if (Cols != Rows) {
			if (ok != 0) *ok = false;
			return *this;
		}
		_CMatrix smat(*this);
		bool ndet;
		uint crow;
		Type mul;
		for (uint i = 0; i < Cols; ++i) {
			ndet = true;
			for (uint j = 0; j < Rows; ++j) if (smat.m[i][j] != 0) ndet = false;
			if (ndet) {
				if (ok != 0) *ok = false;
				return *this;
			}
			for (uint j = 0; j < Cols; ++j) if (smat.m[j][i] != 0) ndet = false;
			if (ndet) {
				if (ok != 0) *ok = false;
				return *this;
			}
		}
		for (uint i = 0; i < Cols; ++i) {
			crow = i;
			while (smat.m[i][i] == Type(0))
				smat.swapRows(i, ++crow);
			for (uint j = i + 1; j < Rows; ++j) {
				mul = smat.m[i][j] / smat.m[i][i];
				for (uint k = i; k < Cols; ++k) smat.m[k][j] -= mul * smat.m[k][i];
			}
			if (i < Cols - 1) {
				if (fabs(smat.m[i+1][i+1]) < Type(1E-100)) {
					if (ok != 0) *ok = false;
					return *this;
				}
			}
		}
		if (ok != 0) *ok = true;
		m = smat.m;
		return *this;
	}

	_CMatrix & invert(bool * ok = 0) {
		if (Cols != Rows) {
			if (ok != 0) *ok = false;
			return *this;
		}
		_CMatrix mtmp = _CMatrix::identity(), smat(*this);
		bool ndet;
		uint crow;
		Type mul, iddiv;
		for (uint i = 0; i < Cols; ++i) {
			ndet = true;
			for (uint j = 0; j < Rows; ++j) if (smat.m[i][j] != 0) ndet = false;
			if (ndet) {
				if (ok != 0) *ok = false;
				return *this;
			}
			for (uint j = 0; j < Cols; ++j) if (smat.m[j][i] != 0) ndet = false;
			if (ndet) {
				if (ok != 0) *ok = false;
				return *this;
			}
		}
		for (uint i = 0; i < Cols; ++i) {
			crow = i;
			while (smat.m[i][i] == Type(0)) {
				++crow;
				smat.swapRows(i, crow);
				mtmp.swapRows(i, crow);
			}
			for (uint j = i + 1; j < Rows; ++j) {
				mul = smat.m[i][j] / smat.m[i][i];
				for (uint k = i; k < Cols; ++k) smat.m[k][j] -= mul * smat.m[k][i];
				for (uint k = 0; k < Cols; ++k) mtmp.m[k][j] -= mul * mtmp.m[k][i];
			}
			//cout << i << endl << smat << endl;
			if (i < Cols - 1) {
				if (fabs(smat.m[i+1][i+1]) < Type(1E-100)) {
					if (ok != 0) *ok = false;
					return *this;
				}
			}
			iddiv = smat.m[i][i];
			for (uint j = i; j < Cols; ++j) smat.m[j][i] /= iddiv;
			for (uint j = 0; j < Cols; ++j) mtmp.m[j][i] /= iddiv;
		}
		for (uint i = Cols - 1; i > 0; --i) {
			for (uint j = 0; j < i; ++j) {
				mul = smat.m[i][j];
				smat.m[i][j] -= mul;
				for (uint k = 0; k < Cols; ++k) mtmp.m[k][j] -= mtmp.m[k][i] * mul;
			}
		}
		if (ok != 0) *ok = true;
		m = mtmp.m;
		return *this;
	}
	_CMatrix inverted(bool * ok = 0) {_CMatrix tm(*this); tm.invert(ok); return tm;}
	_CMatrixI transposed() {_CMatrixI tm; PIMM_FOR_WB(c, r) tm[r][c] = m[c][r]; return tm;}

private:
	void resize(uint cols, uint rows, const Type & new_value = Type()) {m.resize(cols); PIMM_FOR_C(i) m[i].resize(rows, new_value);}
	PIVector<PIVector<Type> > m;

};

template<uint Cols, uint Rows, typename Type>
inline std::ostream & operator <<(std::ostream & s, const PIMathMatrixT<Cols, Rows, Type> & m) {s << '{'; PIMM_FOR_I(c, r) s << m[c][r]; if (c < Cols - 1 || r < Rows - 1) s << ", ";} if (r < Rows - 1) s << endl << ' ';} s << '}'; return s;}

/// Multiply matrices {CR x Rows0} on {Cols1 x CR}, result is {Cols1 x Rows0}
template<uint CR, uint Rows0, uint Cols1, typename Type>
inline PIMathMatrixT<Cols1, Rows0, Type> operator *(const PIMathMatrixT<CR, Rows0, Type> & fm,
													const PIMathMatrixT<Cols1, CR, Type> & sm) {
	PIMathMatrixT<Cols1, Rows0, Type> tm;
	Type t;
	for (uint j = 0; j < Rows0; ++j) {
		for (uint i = 0; i < Cols1; ++i) {
			t = Type(0);
			for (uint k = 0; k < CR; ++k)
				t += fm[k][j] * sm[i][k];
			tm[i][j] = t;
		}
	}
	return tm;
}

/// Multiply matrix {Cols x Rows} on vector {Cols}, result is vector {Rows}
template<uint Cols, uint Rows, typename Type>
inline PIMathVectorT<Rows, Type> operator *(const PIMathMatrixT<Cols, Rows, Type> & fm,
											const PIMathVectorT<Cols, Type> & sv) {
	PIMathVectorT<Rows, Type> tv;
	Type t;
	for (uint i = 0; i < Rows; ++i) {
		t = Type(0);
		for (uint j = 0; j < Cols; ++j)
			t += fm[j][i] * sv[j];
		tv[i] = t;
	}
	return tv;
}

typedef PIMathMatrixT<2u, 2u, int> PIMathMatrixT22i;
typedef PIMathMatrixT<3u, 3u, int> PIMathMatrixT33i;
typedef PIMathMatrixT<4u, 4u, int> PIMathMatrixT44i;
typedef PIMathMatrixT<2u, 2u, double> PIMathMatrixT22d;
typedef PIMathMatrixT<3u, 3u, double> PIMathMatrixT33d;
typedef PIMathMatrixT<4u, 4u, double> PIMathMatrixT44d;


template<typename Type>
class PIMathMatrix;

#undef PIMV_FOR
#undef PIMM_FOR
#undef PIMM_FOR_WB
#undef PIMM_FOR_I
#undef PIMM_FOR_I_WB
#undef PIMM_FOR_C
#undef PIMM_FOR_R

/// Vector

#define PIMV_FOR(v, s) for (uint v = s; v < size_; ++v)

template<typename Type>
class PIMathVector {
	typedef PIMathVector<Type> _CVector;
public:
	PIMathVector(const uint size = 3) {resize(size);}
	PIMathVector(const uint size, Type fval, ...) {resize(size); c[0] = fval; va_list vl; va_start(vl, fval); PIMV_FOR(i, 1) c[i] = va_arg(vl, Type); va_end(vl);}
	PIMathVector(const PIVector<Type> & val) {resize(val.size); PIMV_FOR(i, 0) c[i] = val[i];}
	PIMathVector(const _CVector & st, const _CVector & fn) {resize(st.size()); PIMV_FOR(i, 0) c[i] = fn[i] - st[i];}

	uint size() const {return size_;}
	_CVector & resize(uint size, const Type & new_value = Type()) {size_ = size; c.resize(size, new_value); return *this;}
	_CVector resized(uint size, const Type & new_value = Type()) {_CVector tv = _CVector(*this); tv.resize(size, new_value); return tv;}
	_CVector & fill(const Type & v) {PIMV_FOR(i, 0) c[i] = v; return *this;}
	_CVector & set(Type fval, ...) {c[0] = fval; va_list vl; va_start(vl, fval); PIMV_FOR(i, 1) c[i] = va_arg(vl, Type); va_end(vl); return *this;}
	_CVector & move(const Type & v) {PIMV_FOR(i, 0) c[i] += v; return *this;}
	_CVector & move(const _CVector & v) {PIMV_FOR(i, 0) c[i] += v[i]; return *this;}
	_CVector & move(Type fval, ...) {c[0] += fval; va_list vl; va_start(vl, fval); PIMV_FOR(i, 1) c[i] += va_arg(vl, Type); va_end(vl); return *this;}
	_CVector & swap(uint fe, uint se) {piSwap<Type>(c[fe], c[se]); return *this;}
	Type lengthSqr() const {Type tv(0); PIMV_FOR(i, 0) tv += (c[i] * c[i]); return tv;}
	Type length() const {return sqrt(lengthSqr());}
	Type manhattanLength() const {Type tv(0); PIMV_FOR(i, 0) tv += fabs(c[i]); return tv;}
	Type angleCos(const _CVector & v) const {Type tv = v.length() * length(); return (tv == Type(0) ? Type(0) : ((*this) ^ v) / tv);}
	Type angleSin(const _CVector & v) const {Type tv = angleCos(v); return sqrt(Type(1) - tv * tv);}
	Type angleRad(const _CVector & v) const {return acos(angleCos(v));}
	Type angleDeg(const _CVector & v) const {return acos(angleCos(v)) * rad2deg;}
	_CVector projection(const _CVector & v) {Type tv = v.length(); return (tv == Type(0) ? _CVector() : v * (((*this) ^ v) / tv));}
	_CVector & normalize() {Type tv = length(); if (tv == Type(1)) return *this; PIMV_FOR(i, 0) c[i] /= tv; return *this;}
	_CVector normalized() {_CVector tv(*this); tv.normalize(); return tv;}
	bool isNull() const {PIMV_FOR(i, 0) if (c[i] != Type(0)) return false; return true;}
	bool isOrtho(const _CVector & v) const {return ((*this) ^ v) == Type(0);}

	Type & at(uint index) {return c[index];}
	Type at(uint index) const {return c[index];}
	Type & operator [](uint index) {return c[index];}
	Type operator [](uint index) const {return c[index];}
	void operator =(const _CVector & v) {c = v.c;}
	bool operator ==(const _CVector & v) const {PIMV_FOR(i, 0) if (c[i] != v[i]) return false; return true;}
	bool operator !=(const _CVector & v) const {return !(*this == c);}
	void operator +=(const _CVector & v) {PIMV_FOR(i, 0) c[i] += v[i];}
	void operator -=(const _CVector & v) {PIMV_FOR(i, 0) c[i] -= v[i];}
	void operator *=(const Type & v) {PIMV_FOR(i, 0) c[i] *= v;}
	void operator *=(const _CVector & v) {PIMV_FOR(i, 0) c[i] *= v[i];}
	void operator /=(const Type & v) {PIMV_FOR(i, 0) c[i] /= v;}
	void operator /=(const _CVector & v) {PIMV_FOR(i, 0) c[i] /= v[i];}
	_CVector operator -() {_CVector tv; PIMV_FOR(i, 0) tv[i] = -c[i]; return tv;}
	_CVector operator +(const _CVector & v) {_CVector tv = _CVector(*this); PIMV_FOR(i, 0) tv[i] += v[i]; return tv;}
	_CVector operator -(const _CVector & v) {_CVector tv = _CVector(*this); PIMV_FOR(i, 0) tv[i] -= v[i]; return tv;}
	_CVector operator *(const Type & v) {_CVector tv = _CVector(*this); PIMV_FOR(i, 0) tv[i] *= v; return tv;}
	_CVector operator /(const Type & v) {_CVector tv = _CVector(*this); PIMV_FOR(i, 0) tv[i] /= v; return tv;}
	_CVector operator *(const _CVector & v) {if (size_ < 3) return _CVector(); _CVector tv; tv.fill(Type(1)); tv[0] = c[1]*v[2] - v[1]*c[2]; tv[1] = v[0]*c[2] - c[0]*v[2]; tv[2] = c[0]*v[1] - v[0]*c[1]; return tv;}
	Type operator ^(const _CVector & v) const {Type tv(0); PIMV_FOR(i, 0) tv += c[i] * v[i]; return tv;}

	//inline operator PIMathMatrix<1, Size, Type>() {return PIMathMatrix<1, Size, Type>(c);}
	Type distToLine(const _CVector & lp0, const _CVector & lp1) {
		_CVector a(lp0, lp1), b(lp0, *this), c(lp1, *this);
		Type f = fabs(a[0]*b[1] - a[1]*b[0]) / a.length();//, s = b.length() + c.length() - a.length();
		return f;}

	template<typename Type1>
	PIMathVector turnTo(uint size) {PIMathVector<Type1> tv; uint sz = piMin<uint>(size_, size); for (uint i = 0; i < sz; ++i) tv[i] = c[i]; return tv;}

private:
	uint size_;
	PIVector<Type> c;

};

template<typename Type>
inline std::ostream & operator <<(std::ostream & s, const PIMathVector<Type> & v) {s << '{'; for (uint i = 0; i < v.size(); ++i) {s << v[i]; if (i < v.size() - 1) s << ", ";} s << '}'; return s;}

typedef PIMathVector<int> PIMathVectori;
typedef PIMathVector<double> PIMathVectord;

/// Matrix

#define PIMM_FOR(c, r) for (uint c = 0; c < cols_; ++c) { for (uint r = 0; r < rows_; ++r) {
#define PIMM_FOR_WB(c, r) for (uint c = 0; c < cols_; ++c) for (uint r = 0; r < rows_; ++r) // without brakes
#define PIMM_FOR_I(c, r) for (uint r = 0; r < rows_; ++r) { for (uint c = 0; c < cols_; ++c) {
#define PIMM_FOR_I_WB(c, r) for (uint r = 0; r < rows_; ++r) for (uint c = 0; c < cols_; ++c) // without brakes
#define PIMM_FOR_C(v) for (uint v = 0; v < cols_; ++v)
#define PIMM_FOR_R(v) for (uint v = 0; v < rows_; ++v)

template<typename Type>
class PIMathMatrix {
	typedef PIMathMatrix<Type> _CMatrix;
	typedef PIMathVector<Type> _CMCol;
	typedef PIMathVector<Type> _CMRow;
public:
	PIMathMatrix(const uint cols = 3, const uint rows = 3) {resize(cols, rows);}
	PIMathMatrix(const uint cols, const uint rows, Type fval, ...) {resize(cols, rows); va_list vl; va_start(vl, fval); PIMM_FOR_I_WB(c, r) m[c][r] = (r + c == 0 ? fval : va_arg(vl, Type)); va_end(vl);}
	PIMathMatrix(const uint cols, const uint rows, const PIVector<Type> & val) {resize(cols, rows); int i = 0; PIMM_FOR_I_WB(c, r) m[c][r] = val[i++];}

	static _CMatrix identity(const uint cols_, const uint rows_) {_CMatrix tm(cols_, rows_); PIMM_FOR_WB(c, r) tm.m[c][r] = (c == r ? Type(1) : Type(0)); return tm;}

	uint cols() const {return cols_;}
	uint rows() const {return rows_;}
	_CMCol col(uint index) {_CMCol tv; PIMM_FOR_R(i) tv[i] = m[index][i]; return tv;}
	_CMRow row(uint index) {_CMRow tv; PIMM_FOR_C(i) tv[i] = m[i][index]; return tv;}
	_CMatrix & resize(const uint cols, const uint rows, const Type & new_value = Type()) {cols_ = cols; rows_ = rows; m.resize(cols); PIMM_FOR_C(i) m[i].resize(rows, new_value); return *this;}
	_CMatrix & setCol(uint index, const _CMCol & v) {PIMM_FOR_R(i) m[index][i] = v[i]; return *this;}
	_CMatrix & setRow(uint index, const _CMRow & v) {PIMM_FOR_C(i) m[i][index] = v[i]; return *this;}
	_CMatrix & swapRows(uint r0, uint r1) {Type t; PIMM_FOR_C(i) {t = m[i][r0]; m[i][r0] = m[i][r1]; m[i][r1] = t;} return *this;}
	_CMatrix & swapCols(uint c0, uint c1) {Type t; PIMM_FOR_R(i) {t = m[c0][i]; m[c0][i] = m[c1][i]; m[c1][i] = t;} return *this;}
	_CMatrix & fill(const Type & v) {PIMM_FOR_WB(c, r) m[c][r] = v; return *this;}
	//inline _CMatrix & set(Type fval, ...) {m[0] = fval; va_list vl; va_start(vl, fval); PIMV_FOR(i, 1) m[i] = va_arg(vl, Type); va_end(vl); return *this;}
	//inline void normalize() {Type tv = length(); if (tv == Type(1)) return; PIMV_FOR(i, 0) m[i] /= tv;}
	bool isSquare() const {return cols() == rows();}
	bool isIdentity() const {PIMM_FOR_WB(c, r) if ((c == r) ? m[c][r] != Type(1) : m[c][r] != Type(0)) return false; return true;}
	bool isNull() const {PIMM_FOR_WB(c, r) if (m[c][r] != Type(0)) return false; return true;}

	Type & at(uint col, uint row) {return m[col][row];}
	Type at(uint col, uint row) const {return m[col][row];}
	PIVector<Type> & operator [](uint col) {return m[col];}
	PIVector<Type> operator [](uint col) const {return m[col];}
	void operator =(const _CMatrix & sm) {m = sm.m;}
	bool operator ==(const _CMatrix & sm) const {PIMM_FOR_WB(c, r) if (m[c][r] != sm.m[c][r]) return false; return true;}
	bool operator !=(const _CMatrix & sm) const {return !(*this == sm);}
	void operator +=(const _CMatrix & sm) {PIMM_FOR_WB(c, r) m[c][r] += sm.m[c][r];}
	void operator -=(const _CMatrix & sm) {PIMM_FOR_WB(c, r) m[c][r] -= sm.m[c][r];}
	void operator *=(const Type & v) {PIMM_FOR_WB(c, r) m[c][r] *= v;}
	void operator /=(const Type & v) {PIMM_FOR_WB(c, r) m[c][r] /= v;}
	_CMatrix operator -() {_CMatrix tm(*this); PIMM_FOR_WB(c, r) tm.m[c][r] = -m[c][r]; return tm;}
	_CMatrix operator +(const _CMatrix & sm) {_CMatrix tm(*this); PIMM_FOR_WB(c, r) tm.m[c][r] += sm.m[c][r]; return tm;}
	_CMatrix operator -(const _CMatrix & sm) {_CMatrix tm(*this); PIMM_FOR_WB(c, r) tm.m[c][r] -= sm.m[c][r]; return tm;}
	_CMatrix operator *(const Type & v) {_CMatrix tm(*this); PIMM_FOR_WB(c, r) tm.m[c][r] *= v; return tm;}
	_CMatrix operator /(const Type & v) {_CMatrix tm(*this); PIMM_FOR_WB(c, r) tm.m[c][r] /= v; return tm;}

	_CMatrix & toUpperTriangular(bool * ok = 0) {
		if (cols_ != rows_) {
			if (ok != 0) *ok = false;
			return *this;
		}
		_CMatrix smat(*this);
		bool ndet;
		uint crow;
		Type mul;
		for (uint i = 0; i < cols_; ++i) {
			ndet = true;
			for (uint j = 0; j < rows_; ++j) if (smat.m[i][j] != 0) ndet = false;
			if (ndet) {
				if (ok != 0) *ok = false;
				return *this;
			}
			for (uint j = 0; j < cols_; ++j) if (smat.m[j][i] != 0) ndet = false;
			if (ndet) {
				if (ok != 0) *ok = false;
				return *this;
			}
		}
		for (uint i = 0; i < cols_; ++i) {
			crow = i;
			while (smat.m[i][i] == Type(0))
				smat.swapRows(i, ++crow);
			for (uint j = i + 1; j < rows_; ++j) {
				mul = smat.m[i][j] / smat.m[i][i];
				for (uint k = i; k < cols_; ++k) smat.m[k][j] -= mul * smat.m[k][i];
			}
			if (i < cols_ - 1) {
				if (fabs(smat.m[i+1][i+1]) < Type(1E-100)) {
					if (ok != 0) *ok = false;
					return *this;
				}
			}
		}
		if (ok != 0) *ok = true;
		m = smat.m;
		return *this;
	}

	_CMatrix & invert(bool * ok = 0, _CMCol * sv = 0) {
		if (cols_ != rows_) {
			if (ok != 0) *ok = false;
			return *this;
		}
		_CMatrix mtmp = _CMatrix::identity(cols_, rows_), smat(*this);
		bool ndet;
		uint crow;
		Type mul, iddiv;
		for (uint i = 0; i < cols_; ++i) {
			ndet = true;
			for (uint j = 0; j < rows_; ++j) if (smat.m[i][j] != 0) ndet = false;
			if (ndet) {
				if (ok != 0) *ok = false;
				return *this;
			}
			for (uint j = 0; j < cols_; ++j) if (smat.m[j][i] != 0) ndet = false;
			if (ndet) {
				if (ok != 0) *ok = false;
				return *this;
			}
		}
		for (uint i = 0; i < cols_; ++i) {
			crow = i;
			while (smat.m[i][i] == Type(0)) {
				++crow;
				smat.swapRows(i, crow);
				mtmp.swapRows(i, crow);
				if (sv != 0) sv->swap(i, crow);
			}
			for (uint j = i + 1; j < rows_; ++j) {
				mul = smat.m[i][j] / smat.m[i][i];
				for (uint k = i; k < cols_; ++k) smat.m[k][j] -= mul * smat.m[k][i];
				for (uint k = 0; k < cols_; ++k) mtmp.m[k][j] -= mul * mtmp.m[k][i];
				if (sv != 0) (*sv)[j] -= mul * (*sv)[i];
			}
			//cout << i << endl << smat << endl;
			if (i < cols_ - 1) {
				if (fabs(smat.m[i+1][i+1]) < Type(1E-100)) {
					if (ok != 0) *ok = false;
					return *this;
				}
			}
			iddiv = smat.m[i][i];
			for (uint j = i; j < cols_; ++j) smat.m[j][i] /= iddiv;
			for (uint j = 0; j < cols_; ++j) mtmp.m[j][i] /= iddiv;
			if (sv != 0) (*sv)[i] /= iddiv;
		}
		for (uint i = cols_ - 1; i > 0; --i) {
			for (uint j = 0; j < i; ++j) {
				mul = smat.m[i][j];
				smat.m[i][j] -= mul;
				for (uint k = 0; k < cols_; ++k) mtmp.m[k][j] -= mtmp.m[k][i] * mul;
				if (sv != 0) (*sv)[j] -= mul * (*sv)[i];
			}
		}
		if (ok != 0) *ok = true;
		m = mtmp.m;
		return *this;
	}
	_CMatrix inverted(bool * ok = 0) {_CMatrix tm(*this); tm.invert(ok); return tm;}
	_CMatrix transposed() {_CMatrix tm(rows_, cols_); PIMM_FOR_WB(c, r) tm[r][c] = m[c][r]; return tm;}

private:
	uint cols_, rows_;
	PIVector<PIVector<Type> > m;

};

template<typename Type>
inline std::ostream & operator <<(std::ostream & s, const PIMathMatrix<Type> & m) {s << '{'; for (uint r = 0; r < m.rows(); ++r) { for (uint c = 0; c < m.cols(); ++c) { s << m[c][r]; if (c < m.cols() - 1 || r < m.rows() - 1) s << ", ";} if (r < m.rows() - 1) s << endl << ' ';} s << '}'; return s;}

/// Multiply matrices {CR x Rows0} on {Cols1 x CR}, result is {Cols1 x Rows0}
template<typename Type>
inline PIMathMatrix<Type> operator *(const PIMathMatrix<Type> & fm,
									 const PIMathMatrix<Type> & sm) {
	uint cr = fm.cols(), rows0 = fm.rows(), cols1 = sm.cols();
	PIMathMatrix<Type> tm(cols1, rows0);
	if (fm.cols() != sm.rows()) return tm;
	Type t;
	for (uint j = 0; j < rows0; ++j) {
		for (uint i = 0; i < cols1; ++i) {
			t = Type(0);
			for (uint k = 0; k < cr; ++k)
				t += fm[k][j] * sm[i][k];
			tm[i][j] = t;
		}
	}
	return tm;
}

/// Multiply matrix {Cols x Rows} on vector {Cols}, result is vector {Rows}
template<typename Type>
inline PIMathVector<Type> operator *(const PIMathMatrix<Type> & fm,
									 const PIMathVector<Type> & sv) {
	uint c = fm.cols(), r = fm.rows();
	PIMathVector<Type> tv(r);
	if (c != sv.size()) return tv;
	Type t;
	for (uint i = 0; i < r; ++i) {
		t = Type(0);
		for (uint j = 0; j < c; ++j)
			t += fm[j][i] * sv[j];
		tv[i] = t;
	}
	return tv;
}

typedef PIMathMatrix<int> PIMathMatrixi;
typedef PIMathMatrix<double> PIMathMatrixd;

#undef PIMV_FOR
#undef PIMM_FOR
#undef PIMM_FOR_WB
#undef PIMM_FOR_I
#undef PIMM_FOR_I_WB
#undef PIMM_FOR_C
#undef PIMM_FOR_R


/*
 Fast Fourier Transformation: direct (complement= false)
 and complement (complement = true). 'x' is data source.
 'x' contains 2^T items.
*/
void fft(complexd * x, int T, bool complement);

/// Differential evaluations

struct TransferFunction { // Для задания передаточной функции
	PIVector<double> vector_Bm, vector_An;
};

// Класс, служащий для перевода передаточной функции в систему ОДУ первого порядка
// реализованы след. методы решения дифф. ур-ний:
//	Эйлера
//	Рунге-Кутта 4-го порядка
//	Адамса-Бэшфортса-Моултона 2, 3, 4 порядков
class Solver
{
public:
	enum Method {Global = -1,
				 Eyler_1 = 01,
				 Eyler_2 = 02,
				 EylerKoshi = 03,
				 RungeKutta_4 = 14,
				 AdamsBashfortMoulton_2 = 22,
				 AdamsBashfortMoulton_3 = 23,
				 AdamsBashfortMoulton_4 = 24,
				 PolynomialApproximation_2 = 32,
				 PolynomialApproximation_3 = 33,
				 PolynomialApproximation_4 = 34,
				 PolynomialApproximation_5 = 35
				 };

	Solver() {times.resize(4); step = 0;}

	void solve(double u, double h);
	void fromTF(const TransferFunction & TF);
	void setMethod(Method m) {method = m;}
	void setTime(double time) {times.pop_back(); times.push_front(time);}

	void solveEyler1(double u, double h);
	void solveEyler2(double u, double h);
	void solveRK4(double u, double h);
	void solveABM2(double u, double h);
	void solveABM3(double u, double h);
	void solveABM4(double u, double h);
	void solvePA(double u, double h, uint deg);
	void solvePA2(double u, double h) {if (step > 0) solvePA(u, h, 2); else solveEyler1(u, h);}
	void solvePA3(double u, double h) {if (step > 1) solvePA(u, h, 3); else solvePA2(u, h);}
	void solvePA4(double u, double h) {if (step > 2) solvePA(u, h, 4); else solvePA3(u, h);}
	void solvePA5(double u, double h) {if (step > 3) solvePA(u, h, 5); else solvePA4(u, h);}

	PIMathVectord X;
	static Method method_global;
	static const char methods_desc[];

private:
	void moveF() {for (uint i = F.size() - 1; i > 0; --i) F[i] = F[i - 1];}

	PIMathMatrixd A, M;
	PIMathVectord d, a1, b1;
	PIMathVectord k1, k2, k3, k4, xx;
	PIMathVectord XX, Y, pY;
	PIVector<PIMathVectord> F;
	PIVector<double> times;
	uint size, step;
	Method method;
	double sum, td, ct, lp, dh, t, x1, x0;
	bool ok;

};


#endif // PIMATH_H