Mathematical framework

3 Mathematical framework

3.1 General tools

The macros and functions of this paragraph are defined in pnl/pnl_mathtools.h.

3.1.1 Constants

A few mathematical constants are provided by the library. Most of them are actually already defined in math.h, values.h or limits.h and a few others have been added.

M_E		e¹
M_LOG2E		log ₂e
M_LOG10E		log ₁₀e
M_LN2		log _e2
M_LN10		log _e10
M_PI		π
M_2PI		2π
M_PI_2		π∕2
M_PI_4		π∕4
M_1_PI		1∕π
M_2_PI		2∕π
M_2_SQRTPI		2∕
M_SQRT2PI		sqrt2π
M_SQRT2
M_EULER		γ = lim _n→∞
M_SQRT1_2		1∕
M_1_SQRT2PI		1∕
M_SQRT2_PI
INT_MAX		2147483647
MAX_INT		INT_MAX
DBL_MAX		1.79769313486231470e + 308
DOUBLE_MAX		DBL_MAX
DBL_EPSILON		2.2204460492503131e - 16
PNL_NEGINF		-∞
PNL_POSINF		+∞
PNL_INF		+∞
NAN		Not a Number

3.1.2 A few macros

PNL_IS_ODD (int n)
Description Return 1 if n is odd and 0 otherwise.
PNL_IS_EVEN (int n)
Description Return 1 if n is even and 0 otherwise.
PNL_ALTERNATE (int n)
Description Return (-1)ⁿ.
MIN (x,y)
Description Return the minimum of x and y.
MAX (x,y)
Description Return the maximum of x and y.
ABS (x)
Description Return the absolute value of x.
PNL_SIGN (x)
Description Return the sign of x (-1 if x < 0, 0 otheriwse).
SQR (x)
Description Return x².
CUB (x)
Description Return x³.

3.1.3 Classifying a floating–point number

double pnl_nan ()
Description Return NaN
double pnl_posinf ()
Description Return + infinity
double pnl_neginf ()
Description Return - infinity
int pnl_isnan (double x)
Description Return +1 if x=NaN
int pnl_isinf (double x)
Description Return +1 if x=+Inf, -1 if x=-Inf and 0 otherwise.
int pnl_isfinite (double x)
Description Return 1 if x!=+-Inf

3.1.4 Rouding a floating–point number

int pnl_itrunc (double s)
Description This function is similar to the trunc function (provided by the C library) but the result is typed as an integer instead of a double. Digits may be lost if s exceeds MAX_INT.
long int pnl_ltrunc (double s)
Description This function is similar to the trunc function (provided by the C library) but the result is typed as a long integer instead of a double.
double pnl_trunc (double s)
Description Return the nearest integer not greater than the absolute value of s. This function is part of C99 as trunc.
double pnl_round (double s)
Description Return the integral value nearest to x rounding half-way cases away from zero, regardless of the current rounding direction. This function is part of C99 as round.
int pnl_iround (double s)
Description This function is similar to the round function (provided by the C library) but the result is typed as an integer instead of a double. Digits may be lost if s exceeds MAX_INT.
long int pnl_lround (double s)
Description This function is similar to the round function (provided by the C library) but the result is typed as a long integer instead of a double.

3.1.5 Some standard mathematical functions

double pnl_fact (double x)
Description See pnl_sf_fact
double pnl_lgamma (double x)
Description See pnl_sf_log_gamma
double pnl_tgamma (double x)
Description See pnl_sf_gamma
double pnl_acosh (double x)
Description Compute acosh(x).
double pnl_asinh (double x)
Description Compute asinh(x).
double pnl_atanh (double x)
Description Compute atanh(x).
double pnl_log1p (double x)
Description Compute log(1+x) accurately for small values of x
double pnl_expm1 (double x)
Description Compute exp(x)-1 accurately for small values of x
double pnl_cosm1 (double x)
Description Compute cos(x)-1 accurately for small values of x
double pnl_pow_i (double x, int n)
Description Compute x^n for an integer n.

3.1.6 Comparison of floating–point numbers

int pnl_isequal_rel (double x, double y, double relerr)
Description Compare two floating–point numbers up to a relative precision relerr
int pnl_isequal_abs (double x, double y, double abserr)
Description Compare two floating–point numbers up to an absolute precision abserr
int pnl_isequal (double x, double y, double relerr)
Description Equivalent to pnl_isequal_abs if |x| < 1 and to pnl_isequal_abs otherwise.

3.2 Complex numbers

3.2.1 Overview

The complex type and related functions are defined in the header pnl/pnl_complex.h.

The first native implementation of complex numbers in the C language appeared in C99, which is unfortunately not available on all platforms. For this reason, we provide here an implementation of complex numbers.

typedef struct { 
    double r; /*!< real part */ 
    double i; /*!< imaginary part */ 
} dcomplex;

3.2.2 Constants

CZERO		0 as a complex number
CONE		1 as a complex number
CI		I the unit complex number

3.2.3 Functions

double, double CMPLX (dcomplex z)
Description z.r, z.i
dcomplex Complex (double x, double y)
Description x + i y
dcomplex Complex_polar (double r, double theta)
Description r exp(i theta)
double Creal (dcomplex z)
Description R(z)
double Cimag (dcomplex z)
Description Im(z)
dcomplex Cadd (dcomplex z, dcomplex b)
Description z+b
dcomplex CRadd (dcomplex z, double b)
Description z+b
dcomplex RCadd (double b, dcomplex z)
Description b+z
dcomplex Csub (dcomplex z, dcomplex b)
Description z-b
dcomplex CRsub (dcomplex z, double b)
Description z-b
dcomplex RCsub (double b, dcomplex z)
Description b-z
dcomplex Cminus (dcomplex z)
Description -z
dcomplex Cmul (dcomplex z, dcomplex b)
Description z*b
dcomplex RCmul (double x, dcomplex z)
Description x*z
dcomplex CRmul (dcomplex z, double x)
Description z * x
dcomplex CRdiv (dcomplex z, double x)
Description z/x
dcomplex RCdiv (double x, dcomplex z)
Description x/z
dcomplex Conj (dcomplex z)
Description z
dcomplex Cinv (dcomplex z)
Description 1/z
dcomplex Cdiv (dcomplex z, dcomplex w)
Description z/w
double Csqr_norm (dcomplex z)
Description Re(z)² + im(z)²
double Cabs (dcomplex z)
Description |z|
dcomplex Csqrt (dcomplex z)
Description sqrt(z) , square root (with positive real part)
dcomplex Clog (dcomplex z)
Description log(z)
dcomplex Cexp (dcomplex z)
Description exp(z)
dcomplex CIexp (double t)
Description exp( it )
dcomplex Cpow (dcomplex z, dcomplex w)
Description z^w, power function
dcomplex Cpow_real (dcomplex z, double x)
Description z^x, power function
dcomplex Ccos (dcomplex z)
Description cos(g)
dcomplex Csin (dcomplex z)
Description sin(g)
dcomplex Ctan (dcomplex z)
Description tan(z)
dcomplex Ccotan (dcomplex z)
Description cotan(z)
dcomplex Ccosh (dcomplex z)
Description cosh(g)
dcomplex Csinh (dcomplex z)
Description sinh(g)
dcomplex Ctanh (dcomplex z)
Description tanh(z) =
dcomplex Ccotanh (dcomplex z)
Description cotanh(z) =
double Carg (dcomplex z)
Description arg(z)
dcomplex Ctgamma (dcomplex z)
Description Gamma(z), the Gamma function
dcomplex Clgamma (dcomplex z)
Description log(Gamma (z)), the logarithm of the Gamma function
void Cprintf (dcomplex z)
Description Print a complex number on the standard output
int pnl_complex_isequal_abs (dcomplex x, dcomplex y, double abserr)
Description Test if two complex numbers are equal up to an absolute error abserr.
int pnl_complex_isequal_rel (dcomplex x, dcomplex y, double relerr)
Description Test if two complex numbers are equal up to a relative error relerr.
int pnl_complex_isequal (dcomplex x, dcomplex y, double err)
Description Test if two complex numbers using one of the above functions depending on the magnitude of |y|.

Most algebraic operations on complex numbers are implemented using the following naming for the functions

All these function names begin in C_op_,
The small letters a, b denote two complex numbers whereas d is a real number,
The letter i denotes the multiplication by the pure imagniary number ,
The letter c indicates that the next coming number is conjugated.
The letters p, m denote the two standard operations plus and minus respectively.

For example C_op_idamcb is d ( )
a- b . So functions are :

dcomplex C_op_apib (dcomplex a, dcomplex b)
Description a + b.
dcomplex C_op_apcb (dcomplex a, dcomplex b)
Description a + b.
dcomplex C_op_amcb (dcomplex a, dcomplex b)
Description a -b.
dcomplex C_op_amib (dcomplex a, dcomplex b)
Description a - i b
dcomplex C_op_dapb (double d, dcomplex a, dcomplex b)
Description d(a + b).
dcomplex C_op_damb (double d, dcomplex a, dcomplex b)
Description d(a -b).
dcomplex C_op_dapib (double d, dcomplex a, dcomplex b)
Description d(a + b).
dcomplex C_op_damib (double d, dcomplex a, dcomplex b)
Description d(a -b).
dcomplex C_op_dapcb (double d, dcomplex a, dcomplex b)
Description d.
dcomplex C_op_damcb (double d, dcomplex a, dcomplex b)
Description d.
dcomplex C_op_idapb (double d, dcomplex a, dcomplex b)
Description d.
dcomplex C_op_idamb (double d, dcomplex a, dcomplex b)
Description d.
dcomplex C_op_idapcb (double d, dcomplex a, dcomplex b)
Description d.
dcomplex C_op_idamcb (double d, dcomplex a, dcomplex b)
Description d.

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