path: root/Doc/Elementary Functions/TrigonometricsDependeces.dot

digraph TrigonometricsFunctions {
	node [shape=circle];

//
// -*- Basic Call -*-
//
F77_Call [label="Fortan Call", shape=doublecircle, color=green];
C_Call [label="C Call", shape=doublecircle, color=blue];
LAPACK_Call [label="LAPACK Call", shape=doublecircle, color=red];

//
// -*- Specific LAPACK Call -*-
//
Dlamch_E_Call [label="dlamch('E')", comment="Précision Machine - LAPACK", shape=doublecircle, color=red];
Dlamch_U_Call [label="dlamch('U')", comment="Borne Inférieure - LAPACK", shape=doublecircle, color=red];
Dlamch_O_Call [label="dlamch('O')", comment="Borne Supérieure - LAPACK", shape=doublecircle, color=red];

//
// -*- Functions definition -*-
//

// Cosinus
subgraph clusterCosinus {
	label="Cosine";
	color=yellow;
	style=filled;
	Cos_Real [label="cos(R)"];
	Cos_Complex [label="cos(C)"];
}
subgraph clusterHyperbolicCosinus {
	label="Hyperbolic Cosine";
	color="wheat";
	style=filled;
	Cos_Hyperbolic_Real [label="ch(R)"];
	Cos_Hyperbolic_Complex [label="ch(C)"];
}

// ArcCosinus
subgraph clusterArcCosinus {
	label="Inverse Cosine";
	color=springgreen;
	style=filled;
	ACos_Real [label="acos(R)"];
	ACos_Complex [label="acos(C)"];
}
subgraph clusterArcHyperbolicCosinus {
	label="Inverse Hyperbolic Cosine";
	color=slategray;
	style=filled;
	ACos_Hyperbolic_Real [label="ach(R)"];
	ACos_Hyperbolic_Complex [label="ach(C)"];
}

// Sinus
subgraph clusterSinus {
	label="Sine";
	color="violetred";
	style=filled;
	Sin_Real [label="sin(R)"];
	Sin_Complex [label="sin(C)"];
}
subgraph clusterHyperbolicSinus {
	label="Hyperbolic Sine";
	color=skyblue;
	style=filled;
	Sin_Hyperbolic_Real [label="sh(R)"];
	Sin_Hyperbolic_Complex [label="sh(C)"];
}


// ArcSinus
subgraph clusterArcSinus {
	label="Inverse Sine";
	color=slateblue;
	style=filled;
	ASin_Real [label="asin(R)"];
	ASin_Complex [label="asin(C)"];
}
subgraph clusterArcHyperbolicSinus {
	label="Inverse Hyperbolic Sine";
	color=violet;
	style=filled;
	ASin_Hyperbolic_Real [label="ash(R)"];
	ASin_Hyperbolic_Complex [label="ash(C)"];
}


// Tangeante
subgraph clusterTan {
	label="Tan";
	color=turquoise;
	style=filled;
	Tan_Real [label="tan(R)"];
	Tan_Complex [label="tan(C)", comment="wtan"];
}
subgraph clusterHyperbolicTan {
	label="Hyperbolic Tan";
	color=tomato;
	style=filled;
	Tan_Hyperbolic_Real [label="tanh(R)"];
	Tan_Hyperbolic_Complex [label="tanh(C)"];
}


// ArcTangeante
subgraph clusterArcTan {
	label="Inverse Tan";
	color=limegreen;
	style=filled;
	ATan_Real [label="atan(R)"];
	ATan_Complex [label="atan(C)", comment="watan"];
}
subgraph clusterArcHyperbolicTan {
	label="Inverse Hyperbolic Tan";
	color=mediumorchid;
	style=filled;
	ATan_Hyperbolic_Real [label="atanh(R)"];
	ATan_Hyperbolic_Complex [label="atanh(C)"];
}


// ArcTaneante2
ATan2_Real [label="atan2(R)"];

// Exponentielle
subgraph clusterExponential {
	label="Exponential";
	color=thistle;
	style=filled;
	Exp_Real [label="exp(R)"];
	Exp_Complex [label="exp(C)"];
}

// Log
subgraph clusterLog {
	label="Log";
	color=tan;
	style=filled;
	Log_Real [label="log(R)"];
	Log_Positive_Real [label="log(R+)"];
	Log_Negative_Real [label="log(R-)"];
	Log_Complex [label="log(C)"];
}

// Log1p
Log1p_Real [label="log1p(R)"];

// Racine Carrée
subgraph clusterSquare {
	label="Square Root";
	color=steelblue;
	style=filled;
	Sqrt_Real [label="sqrt(R)"];
	Sqrt_Positive_Real [label="sqrt(R+)"];
	Sqrt_Negative_Real [label="sqrt(R-)"];
	Sqrt_Complex [label="sqrt(C)", comment="wsqrt"];
}

// Partie Imaginaire
Imag_Complex [label="imag(C)"];

// Valeur Absolue
Abs_Real [label="abs(R)", comment="|R|"];

// Signe
Sign_Real [label="sign(R)"];

// Pythagore
Pythag_Real [label="pythag(R)"];


//
// -*- Functions call links -*-
//
subgraph clusterLAPACK {
	style=filled;
	color=lightsteelblue;
	label="LAPACK";
	// -*- DLAMCH -*-
	Dlamch_E_Call -> LAPACK_Call;
	Dlamch_U_Call -> LAPACK_Call;
	Dlamch_O_Call -> LAPACK_Call;
}


// -*- COS -*-
//
// cos(a+ib) = cos(a).ch(b) - i.sin(a).sh(b)
Cos_Complex  -> {
	Cos_Real
	Sin_Real
	Cos_Hyperbolic_Real
	Sin_Hyperbolic_Real
	};

// Call a cosinus function in F77 math lib
Cos_Real -> F77_Call;


// -*- ACOS -*-
//
// acos(a+ib) = see $SCIHOME/modules/elementaries_functions/src/fortran/wacos.f
ACos_Complex -> {
	Dlamch_O_Call
	Dlamch_E_Call
	Dlamch_U_Call
	Sqrt_Real
	Abs_Real
	ACos_Real
	ATan_Real
	Log1p_Real
	Log_Real
	Sign_Real
	};

// Call an inverse cosine function in F77 math lib
ACos_Real -> F77_Call;


// -*- COSH -*-
//
// ch(z) = cos(i.z)
Cos_Hyperbolic_Complex -> {
	Cos_Complex
	};

// ch(x) = 1/2 . exp(|x|) + exp(-|x|)
Cos_Hyperbolic_Real -> {
	Exp_Real
	Abs_Real
	};


// -*- ACOSH -*-
//
// acosh(z) = sign(-imag(acos(z)) i acos(z)
ACos_Hyperbolic_Complex -> {
	Imag_Complex
	ACos_Complex
};
// acosh(z) = sign(-imag(acos(z)) i acos(
ACos_Hyperbolic_Real -> {
	Imag_Complex
	ACos_Complex
	Min_Real
};


// -*- SIN -*-
//
// sin(a+ib) = sin(a).ch(b) + i.cos(a).sh(b)
Sin_Complex -> {
	Cos_Real
	Sin_Real
	Cos_Hyperbolic_Real
	Sin_Hyperbolic_Real
	};

// Call a sinus function in F77 math lib
Sin_Real -> F77_Call;


// -*- ASIN -*-
//
// asin(a+ib) = see $SCIHOME/modules/elementaries_functions/src/fortran/wasin.f
ASin_Complex -> {
	Dlamch_O_Call
	Dlamch_U_Call
	Dlamch_E_Call
	Abs_Real
	Sqrt_Real
	ASin_Real
	ATan_Real
	Log1p_Real
	Log_Real
	Sign_Real
};

// Call an inverse sine function in F77 math lib
ASin_Real -> F77_Call;


// -*- SINH -*-
//
// sh(z) = -i.sin(i.z)
Sin_Hyperbolic_Complex -> {
	Sin_Complex
	};

// sh(x) = imag(sin(i.x))
Sin_Hyperbolic_Real -> {
	Sin_Complex
	Imag_Complex
	};


// -*- ASINH -*-
//
// asinh(z) = -i asin(i z)
ASin_Hyperbolic_Complex -> {
	ASin_Complex
};

// asinh(z) = -i asin(i z)
ASin_Hyperbolic_Real -> {
	ASin_Complex
	Imag_Complex
};


// -*- TAN -*-
//
// tan(a+ib) = x_r + i.x_i
//
// x_r= 1/2 sin(2.a) / d
//
// x_i = | sh(2.b) / 2.d	si |b| <= L
//	 | sign(a)
//
// d = cos(a)^2 + sh(b)^2
//
// L = 1 + log(2 / sqrt(dlamch('e')))
Tan_Complex -> {
	Cos_Real
	Sin_Hyperbolic_Real
	Log_Real
	Sqrt_Real
	Dlamch_E_Call
	Abs_Real
	Sign_Real
	};

// Call a tangeante function in F77 math lib
Tan_Real -> F77_Call;


//
// -*- ATAN -*-
//
ATan_Complex -> {
	Dlamch_O_Call
	ATan_Real
	Abs_Real
	Sign_Real
	Log_Real
	ATan2_Real
};

// Call an inverse tangeant function in F77 math lib
ATan_Real -> F77_Call;


// -*- TANH -*-
//
// tanh(z) = -i.tan(i.z)
Tan_Hyperbolic_Complex -> {
	Tan_Complex
	};

// tanh(x) = imag(tan(i.x))
Tan_Hyperbolic_Real -> {
	Tan_Complex
	Imag_Complex
	};

// -*- ATANH -*-
//
// atanh(z) = i atan(-i z)
ATan_Hyperbolic_Complex -> {
	ATan_Complex
	};

// atanh(x) = -imag(atan(i.x)) | i.atan(-i.x)
ATan_Hyperbolic_Real -> {
	ATan_Complex
	Imag_Complex
	};


// -*- LOG -*-
//
// log(a+ib) = x_r + i.x_i
//
// Constantes :
//		L_inf = sqrt(dlamch('U'))
//		L_sup = sqrt(R_max / 2)
//		R_max = dlamch('O')
//		t = pythag(|a|,|b|)
//		r = |b|/|a|
//
// x_i = atan2(b, a)
//
// x_r = | 1/2 logp1((|a|-1)(|a|+1) + |b|^2)		si 1/2 <= |a| <= sqrt(2)
//	 | 1/2 log(|a|^2 + |b|^2)			si L_inf < |b| && |a| < L_sup
//	 | |a|						si |a| > R_max
//	 | log(t)					si t <= R_max
//	 | log(|a|) + 1/2 logp1(r^2)			sinon
Log_Complex -> {
	Sqrt_Real
	Dlamch_U_Call
	Dlamch_O_Call
	Pythag_Real
	Log1p_Real
	Log_Real
	Abs_Real
	Pythag_Real
	ATan2_Real
	};

// Separate positive and negative case
Log_Real -> {
	Log_Positive_Real
	Log_Negative_Real
	};

// Call another log function, treat it as complex
Log_Negative_Real -> Log_Complex;

// Call a log function in F77 math lib
Log_Positive_Real -> F77_Call;

// Call a log1p function in F77 math lib
Log1p_Real -> F77_Call;


// -*- SQRT -*-
//
// sqrt(a+ib) = x_r + i.x_i
//
// Constantes :
//		Rmax = dlamch('O')
//		BRmin = 2.dlamch('U')
//		t = sqrt(2.|a| + pythag(a,b))
//
// (1) a = 0
//~~~~~~~~~~~~
// x_r = | sqrt(|b| / 2)		si |b| <= BRmin
//	 | sqrt(|b|).sqrt(1/2)
//
// x_i = sign(b).x_r
//
// (2) |a| >= BRmax && |b| >= BRmax
//~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
//  - Si t > Rmax
//	a = a / 16
//      b = b / 16
//      t = sqrt(2.|a| + pythag(a,b))
//	x_r = | 2.t		si a >= 0
//	      | 4.|b| / t
//	x_i = | 4.b / t		si a >= 0
//	      | 2.sign(b).t
//
//  - Sinon
//
//	x_r = | t / 2		si a >= 0
//	      | |b| / t
//
//	x_i = | b / t		si a >= 0
//	      | sign(b) / 2.t
//
// (3) Tous les cas pourris
//~~~~~~~~~~~~~~~~~~~~~~~~~~
//
// x_r = | a + b		si a is NaN && b is NaN
//	 | |b|			si |b| > Rmax
//	 | 0			si a < -Rmax
//	 | a
//
// x_i = | a + b		si a is NaN && b is NaN
//	 | b			si |b| > Rmax
//	 | sign(b).|a|		si a < -Rmax
//	 | 0
//
Sqrt_Complex -> {
	Abs_Real
	Sqrt_Real
	Sign_Real
	Pythag_Real
	Dlamch_U_Call
	Dlamch_O_Call
	};

// Separate positive and negative case
Sqrt_Real -> {
	Sqrt_Positive_Real
	Sqrt_Negative_Real
	};

// Call another sqrt function, treat it as complex
Sqrt_Negative_Real -> Sqrt_Complex;

// Call a sqrt function in F77 math lib
Sqrt_Positive_Real -> F77_Call;

// -*- PYTHAG -*-
//
// pythag(a,b) = sqrt(a^2 + b^2)
//
//
Pythag_Real -> {
	Sqrt_Real
	Dlamch_O_Call
	};



// -*- SIGN -*-
//
// Call a sign function in F77 math lib
Sign_Real -> F77_Call;


// -*- ABS -*-
//
// Call a abs function in F77 math lib
Abs_Real -> F77_Call;


// -*- EXP -*-
//
// exp(a+i.b) = exp(a).cos(b) + i.exp(a)sin(b)
Exp_Complex -> {
	Exp_Real
	Cos_Real
	Sin_Real
	};

// Call a exp function in F77 math lib
Exp_Real -> F77_Call;


// -*- ATAN2 -*-
//
// Call atan2 function in C math lib
ATan2_Real -> C_Call;


// -*- IMAG -*-
//
// Call a imag function in F77 math lib
Imag_Complex -> F77_Call;

}