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This chapter describes basic mathematical functions. Some of these
functions are present in system libraries, but the alternative versions
given here can be used as a substitute when the system functions are not
available.
The functions and macros described in this chapter are defined in the
header file 'gsl_math.h'.
The library ensures that the standard BSD mathematical constants
are defined. For reference here is a list of the constants.
M_E
-
The base of exponentials, e
M_LOG2E
-
The base-2 logarithm of e, \log_2 (e)
M_LOG10E
-
The base-10 logarithm of e,
\log_10 (e)
M_SQRT2
-
The square root of two, \sqrt 2
M_SQRT1_2
-
The square root of one-half,
\sqrt{1/2}
M_SQRT3
-
The square root of three, \sqrt 3
M_PI
-
The constant pi, \pi
M_PI_2
-
Pi divided by two, \pi/2
M_PI_4
-
Pi divided by four, \pi/4
M_SQRTPI
-
The square root of pi, \sqrt\pi
M_2_SQRTPI
-
Two divided by the square root of pi, 2/\sqrt\pi
M_1_PI
-
The reciprocal of pi, 1/\pi
M_2_PI
-
Twice the reciprocal of pi, 2/\pi
M_LN10
-
The natural logarithm of ten, \ln(10)
M_LN2
-
The natural logarithm of two, \ln(2)
M_LNPI
-
The natural logarithm of pi, \ln(\pi)
M_EULER
-
Euler's constant, \gamma
- Macro: GSL_POSINF
-
This macro contains the IEEE representation of positive infinity,
+\infty. It is computed from the expression
+1.0/0.0.
- Macro: GSL_NEGINF
-
This macro contains the IEEE representation of negative infinity,
-\infty. It is computed from the expression
-1.0/0.0.
- Macro: GSL_NAN
-
This macro contains the IEEE representation of the Not-a-Number symbol,
NaN. It is computed from the ratio 0.0/0.0.
- Function: int gsl_isnan (const double x)
-
This function returns 1 if x is not-a-number.
- Function: int gsl_isinf (const double x)
-
This function returns +1 if x is positive infinity,
-1 if x is negative infinity and 0 otherwise.
- Function: int gsl_finite (const double x)
-
This function returns 1 if x is a real number, and 0 if it is
infinite or not-a-number.
The following routines provide portable implementations of functions
found in the BSD math library. When native versions are not available
the functions described here can be used instead. The substitution can
be made automatically if you use autoconf to compile your
application (see section Portability functions).
- Function: double gsl_log1p (const double x)
-
This function computes the value of \log(1+x) in a way that is
accurate for small x. It provides an alternative to the BSD math
function
log1p(x).
- Function: double gsl_expm1 (const double x)
-
This function computes the value of \exp(x)-1 in a way that is
accurate for small x. It provides an alternative to the BSD math
function
expm1(x).
- Function: double gsl_hypot (const double x, const double y)
-
This function computes the value of
\sqrt{x^2 + y^2} in a way that avoids overflow. It provides an
alternative to the BSD math function
hypot(x,y).
- Function: double gsl_acosh (const double x)
-
This function computes the value of \arccosh(x). It provides an
alternative to the standard math function
acosh(x).
- Function: double gsl_asinh (const double x)
-
This function computes the value of \arcsinh(x). It provides an
alternative to the standard math function
asinh(x).
- Function: double gsl_atanh (const double x)
-
This function computes the value of \arctanh(x). It provides an
alternative to the standard math function
atanh(x).
- Function: double gsl_ldexp (double x, int e)
-
This function computes the value of x * 2^e. It provides an
alternative to the standard math function
ldexp(x).
- Function: double gsl_frexp (double x, int * e)
-
This function splits the number x into its normalized fraction
f and exponent e, such that x = f * 2^e and
0.5 <= f < 1. Ihe function returns f and stores the
exponent in e. If x is zero, both f and e
are set to zero. This function provides an alternative to the standard
math function
frexp(x, e).
A common complaint about the standard C library is its lack of a
function for calculating (small) integer powers. GSL provides a simple
functions to fill this gap. For reasons of efficiency, these functions
do not check for overflow or underflow conditions.
- Function: double gsl_pow_int (double x, int n)
-
This routine computes the power x^n for integer n. The
power is computed efficiently -- for example, x^8 is computed as
((x^2)^2)^2, requiring only 3 multiplications. A version of this
function which also computes the numerical error in the result is
available as
gsl_sf_pow_int_e.
- Function: double gsl_pow_2 (const double x)
-
- Function: double gsl_pow_3 (const double x)
-
- Function: double gsl_pow_4 (const double x)
-
- Function: double gsl_pow_5 (const double x)
-
- Function: double gsl_pow_6 (const double x)
-
- Function: double gsl_pow_7 (const double x)
-
- Function: double gsl_pow_8 (const double x)
-
- Function: double gsl_pow_9 (const double x)
-
These functions can be used to compute small integer powers x^2,
x^3, etc. efficiently. The functions will be inlined when
possible so that use of these functions should be as efficient as
explicitly writing the corresponding product expression.
#include <gsl/gsl_math.h>
double y = gsl_pow_4 (3.141) /* compute 3.141**4 */
- Macro: GSL_SIGN (x)
-
This macro returns the sign of x. It is defined as
((x) >= 0
? 1 : -1). Note that with this definition the sign of zero is positive
(regardless of its IEEE sign bit).
- Macro: GSL_IS_ODD (n)
-
This macro evaluates to 1 if n is odd and 0 if n is
even. The argument n must be of integer type.
- Macro: GSL_IS_EVEN (n)
-
This macro is the opposite of
GSL_IS_ODD(n). It evaluates to 1 if
n is even and 0 if n is odd. The argument n must be of
integer type.
- Macro: GSL_MAX (a, b)
-
This macro returns the maximum of a and b. It is defined as
((a) > (b) ? (a):(b)).
- Macro: GSL_MIN (a, b)
-
This macro returns the minimum of a and b. It is defined as
((a) < (b) ? (a):(b)).
- Function: extern inline double GSL_MAX_DBL (double a, double b)
-
This function returns the maximum of the double precision numbers
a and b using an inline function. The use of a function
allows for type checking of the arguments as an extra safety feature. On
platforms where inline functions are not available the macro
GSL_MAX will be automatically substituted.
- Function: extern inline double GSL_MIN_DBL (double a, double b)
-
This function returns the minimum of the double precision numbers
a and b using an inline function. The use of a function
allows for type checking of the arguments as an extra safety feature. On
platforms where inline functions are not available the macro
GSL_MIN will be automatically substituted.
- Function: extern inline int GSL_MAX_INT (int a, int b)
-
- Function: extern inline int GSL_MIN_INT (int a, int b)
-
These functions return the maximum or minimum of the integers a
and b using an inline function. On platforms where inline
functions are not available the macros
GSL_MAX or GSL_MIN
will be automatically substituted.
- Function: extern inline long double GSL_MAX_LDBL (long double a, long double b)
-
- Function: extern inline long double GSL_MIN_LDBL (long double a, long double b)
-
These functions return the maximum or minimum of the long doubles a
and b using an inline function. On platforms where inline
functions are not available the macros
GSL_MAX or GSL_MIN
will be automatically substituted.
It is sometimes useful to be able to compare two floating point numbers
approximately, to allow for rounding and truncation errors. The following
function implements the approximate floating-point comparison algorithm
proposed by D.E. Knuth in Section 4.2.2 of Seminumerical
Algorithms (3rd edition).
- Function: int gsl_fcmp (double x, double y, double epsilon)
-
This function determines whether x and y are approximately
equal to a relative accuracy epsilon.
The relative accuracy is measured using an interval of size 2
\delta, where \delta = 2^k \epsilon and k is the
maximum base-2 exponent of x and y as computed by the
function frexp().
If x and y lie within this interval, they are considered
approximately equal and the function returns 0. Otherwise if x <
y, the function returns -1, or if x > y, the function returns
+1.
The implementation is based on the package fcmp by T.C. Belding.
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