.. index:: single: elementary functions single: mathematical functions, elementary ********************** Mathematical Functions ********************** 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 :file:gsl_math.h. .. index:: single: mathematical constants, defined as macros single: numerical constants, defined as macros single: constants, mathematical (defined as macros) single: macros for mathematical constants Mathematical Constants ====================== The library ensures that the standard BSD mathematical constants are defined. For reference, here is a list of the constants: .. index:: single: e, defined as a macro single: pi, defined as a macro single: Euler's constant, defined as a macro ===================== =================================== :macro:M_E The base of exponentials, :math:e :macro:M_LOG2E The base-2 logarithm of :math:e, :math:\log_2 (e) :macro:M_LOG10E The base-10 logarithm of :math:e, :math:\log_{10} (e) :macro:M_SQRT2 The square root of two, :math:\sqrt 2 :macro:M_SQRT1_2 The square root of one-half, :math:\sqrt{1/2} :macro:M_SQRT3 The square root of three, :math:\sqrt 3 :macro:M_PI The constant pi, :math:\pi :macro:M_PI_2 Pi divided by two, :math:\pi/2 :macro:M_PI_4 Pi divided by four, :math:\pi/4 :macro:M_SQRTPI The square root of pi, :math:\sqrt\pi :macro:M_2_SQRTPI Two divided by the square root of pi, :math:2/\sqrt\pi :macro:M_1_PI The reciprocal of pi, :math:1/\pi :macro:M_2_PI Twice the reciprocal of pi, :math:2/\pi :macro:M_LN10 The natural logarithm of ten, :math:\ln(10) :macro:M_LN2 The natural logarithm of two, :math:\ln(2) :macro:M_LNPI The natural logarithm of pi, :math:\ln(\pi) :macro:M_EULER Euler's constant, :math:\gamma ===================== =================================== .. index:: single: infinity, defined as a macro single: IEEE infinity, defined as a macro Infinities and Not-a-number =========================== .. macro:: GSL_POSINF This macro contains the IEEE representation of positive infinity, :math:+\infty. It is computed from the expression :code:+1.0/0.0. .. macro:: GSL_NEGINF This macro contains the IEEE representation of negative infinity, :math:-\infty. It is computed from the expression :code:-1.0/0.0. .. index:: single: NaN, defined as a macro single: Not-a-number, defined as a macro single: IEEE NaN, defined as a macro .. macro:: GSL_NAN This macro contains the IEEE representation of the Not-a-Number symbol, :code:NaN. It is computed from the ratio :code:0.0/0.0. .. function:: int gsl_isnan (const double x) This function returns 1 if :data:x is not-a-number. .. function:: int gsl_isinf (const double x) This function returns :math:+1 if :data:x is positive infinity, :math:-1 if :data:x is negative infinity and 0 otherwise. [#f1]_ .. function:: int gsl_finite (const double x) This function returns 1 if :data:x is a real number, and 0 if it is infinite or not-a-number. Elementary Functions ==================== 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 :code:autoconf to compile your application (see :ref:portability-functions). .. index:: single: log1p single: logarithm, computed accurately near 1 .. function:: double gsl_log1p (const double x) This function computes the value of :math:\log(1+x) in a way that is accurate for small :data:x. It provides an alternative to the BSD math function :code:log1p(x). .. index:: single: expm1 single: exponential, difference from 1 computed accurately .. function:: double gsl_expm1 (const double x) This function computes the value of :math:\exp(x)-1 in a way that is accurate for small :data:x. It provides an alternative to the BSD math function :code:expm1(x). .. index:: single: hypot single: euclidean distance function, hypot single: length, computed accurately using hypot .. function:: double gsl_hypot (const double x, const double y) This function computes the value of :math:\sqrt{x^2 + y^2} in a way that avoids overflow. It provides an alternative to the BSD math function :code:hypot(x,y). .. index:: single: euclidean distance function, hypot3 single: length, computed accurately using hypot3 .. function:: double gsl_hypot3 (const double x, const double y, const double z) This function computes the value of :math:\sqrt{x^2 + y^2 + z^2} in a way that avoids overflow. .. index:: single: acosh single: hyperbolic cosine, inverse single: inverse hyperbolic cosine .. function:: double gsl_acosh (const double x) This function computes the value of :math:\arccosh{(x)}. It provides an alternative to the standard math function :code:acosh(x). .. index:: single: asinh single: hyperbolic sine, inverse single: inverse hyperbolic sine .. function:: double gsl_asinh (const double x) This function computes the value of :math:\arcsinh{(x)}. It provides an alternative to the standard math function :code:asinh(x). .. index:: single: atanh single: hyperbolic tangent, inverse single: inverse hyperbolic tangent .. function:: double gsl_atanh (const double x) This function computes the value of :math:\arctanh{(x)}. It provides an alternative to the standard math function :code:atanh(x). .. index:: ldexp .. function:: double gsl_ldexp (double x, int e) This function computes the value of :math:x * 2^e. It provides an alternative to the standard math function :code:ldexp(x,e). .. index:: frexp .. function:: double gsl_frexp (double x, int * e) This function splits the number :data:x into its normalized fraction :math:f and exponent :math:e, such that :math:x = f * 2^e and :math:0.5 <= f < 1. The function returns :math:f and stores the exponent in :math:e. If :math:x is zero, both :math:f and :math:e are set to zero. This function provides an alternative to the standard math function :code:frexp(x, e). Small integer powers ==================== A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL provides some 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) double gsl_pow_uint (double x, unsigned int n) These routines computes the power :math:x^n for integer :data:n. The power is computed efficiently---for example, :math:x^8 is computed as :math:((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 :func:gsl_sf_pow_int_e. .. function:: double gsl_pow_2 (const double x) double gsl_pow_3 (const double x) double gsl_pow_4 (const double x) double gsl_pow_5 (const double x) double gsl_pow_6 (const double x) double gsl_pow_7 (const double x) double gsl_pow_8 (const double x) double gsl_pow_9 (const double x) These functions can be used to compute small integer powers :math:x^2, :math:x^3, etc. efficiently. The functions will be inlined when :macro:HAVE_INLINE is defined, so that use of these functions should be as efficient as explicitly writing the corresponding product expression:: #include double y = gsl_pow_4 (3.141) /* compute 3.141**4 */ Testing the Sign of Numbers =========================== .. macro:: GSL_SIGN (x) This macro returns the sign of :data:x. It is defined as :code:((x) >= 0 ? 1 : -1). Note that with this definition the sign of zero is positive (regardless of its IEEE sign bit). Testing for Odd and Even Numbers ================================ .. macro:: GSL_IS_ODD (n) This macro evaluates to 1 if :data:n is odd and 0 if :data:n is even. The argument :data:n must be of integer type. .. macro:: GSL_IS_EVEN (n) This macro is the opposite of :macro:GSL_IS_ODD. It evaluates to 1 if :data:n is even and 0 if :data:n is odd. The argument :data:n must be of integer type. Maximum and Minimum functions ============================= Note that the following macros perform multiple evaluations of their arguments, so they should not be used with arguments that have side effects (such as a call to a random number generator). .. index:: maximum of two numbers .. macro:: GSL_MAX (a, b) This macro returns the maximum of :data:a and :data:b. It is defined as :code:((a) > (b) ? (a):(b)). .. index:: minimum of two numbers .. macro:: GSL_MIN (a, b) This macro returns the minimum of :data:a and :data:b. It is defined as :code:((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 :data:a and :data: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 :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 :data:a and :data: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 :macro:GSL_MIN will be automatically substituted. .. function:: extern inline int GSL_MAX_INT (int a, int b) extern inline int GSL_MIN_INT (int a, int b) These functions return the maximum or minimum of the integers :data:a and :data:b using an inline function. On platforms where inline functions are not available the macros :macro:GSL_MAX or :macro:GSL_MIN will be automatically substituted. .. function:: extern inline long double GSL_MAX_LDBL (long double a, long double b) extern inline long double GSL_MIN_LDBL (long double a, long double b) These functions return the maximum or minimum of the long doubles :data:a and :data:b using an inline function. On platforms where inline functions are not available the macros :macro:GSL_MAX or :macro:GSL_MIN will be automatically substituted. Approximate Comparison of Floating Point Numbers ================================================ 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). .. index:: single: approximate comparison of floating point numbers single: safe comparison of floating point numbers single: floating point numbers, approximate comparison .. function:: int gsl_fcmp (double x, double y, double epsilon) This function determines whether :data:x and :data:y are approximately equal to a relative accuracy :data:epsilon. The relative accuracy is measured using an interval of size :math:2 \delta, where :math:\delta = 2^k \epsilon and :math:k is the maximum base-2 exponent of :math:x and :math:y as computed by the function :func:frexp. If :math:x and :math:y lie within this interval, they are considered approximately equal and the function returns 0. Otherwise if :math:x < y, the function returns :math:-1, or if :math:x > y, the function returns :math:+1. Note that :math:x and :math:y are compared to relative accuracy, so this function is not suitable for testing whether a value is approximately zero. The implementation is based on the package :code:fcmp by T.C. Belding. .. rubric:: Footnotes .. [#f1] Note that the C99 standard only requires the system :func:isinf function to return a non-zero value, without the sign of the infinity. The implementation in some earlier versions of GSL used the system :func:isinf function and may have this behavior on some platforms. Therefore, it is advisable to test the sign of :data:x separately, if needed, rather than relying the sign of the return value from :func:gsl_isinf().