Additional mathematical routines.

This document describes the math unit. The math unit was initially written by Florian Klaempfl. It provides mathematical functions which aren't covered by the system unit.

This chapter starts out with a definition of all types and constants that are defined, after which an overview is presented of the available functions, grouped by category, and the last part contains a complete explanation of each function.

The following things must be taken into account when using this unit:

This unit is compiled in Object Pascal mode so all integers are 32 bit.
Some overloaded functions exist for data arrays of integers and floats. When using the address operator (@) to pass an array of data to such a function, make sure the address is typecasted to the right type, or turn on the 'typed address operator' feature. failing to do so, will cause the compiler not be able to decide which function you want to call.

Min/max determination

Functions to determine the minimum or maximum of numbers:

	Name	Description
	Maximum of 2 values
	Maximum of an array of integer values
	Maximum of an array of values
	Minimum of 2 values
	Minimum of an array of integer values
	Minimum of an array of values

Angle unit conversion

Routines to convert angles between different angle units.

	Name	Description
	convert cycles to radians
	convert degrees to grads
	convert degrees to radians
	convert grads to degrees
	convert grads to radians
	convert radians to cycles
	convert radians to degrees
	convert radians to grads

Trigoniometric functions

	Name	Description
	calculate reverse cosine
	calculate reverse sine
	calculate reverse tangent
	calculate cotangent
	calculate sine and cosine
	calculate tangent

Hyperbolic functions

	Name	Description
	caculate reverse hyperbolic cosine
	caculate reverse hyperbolic sine
	caculate reverse hyperbolic tangent
	calculate hyperbolic cosine
	calculate hyperbolic sine
	calculate hyperbolic tangent

Exponential and logarithmic functions

	Name	Description
	Raise float to integer power
	Calculate $2^p x$
	calculate `log(x+1)`
	calculate 10-base log
	calculate 2-base log
	calculate N-base log
	raise float to arbitrary power

Number converting

	Name	Description
	Round to infinity
	Round to minus infinity
	Return mantissa and exponent

Statistical functions

	Name	Description
	Mean of values
	Mean and standard deviation of values
	Moments, skew and kurtosis
	Population standarddeviation
	Population variance
	Gaussian distributed randum value
	Standard deviation
	Sum of values
	Sum of squared values
	Sum of values and squared values
	Total variance of values
	variance of values

Geometrical functions

	Name	Description
	Hypotenuse of triangle
	Euclidian norm

Float type used in all calls All calculations are done with the Float type. This allows to recompile the unit with a different float type to obtain a desired precision. The pointer type is used in functions that accept an array of values of arbitrary length. Pointer to type. Type used in financial (interest) calculations. End of period. Start of period. Exception raised when invalid arguments are passed to a function. Arccos returns the inverse cosine of its argument x. The argument x should lie between -1 and 1 (borders included). If the argument x is not in the allowed range, an EInvalidArgument exception is raised. Arcosh returns the inverse hyperbolic cosine of its argument x. The argument x should be larger than 1. The arccosh variant of this function is supplied for Delphi compatibility. If the argument x is not in the allowed range, an EInvalidArgument exception is raised. , Arcsin returns the inverse sine of its argument x. The argument x should lie between -1 and 1. If the argument x is not in the allowed range, an EInvalidArgument exception is raised. arctan2 calculates arctan(y/x), and returns an angle in the correct quadrant. The returned angle will be in the range $-\pi$ to $\pi$ radians. The values of x and y must be between -2\^{}64 and 2\^{}64, moreover x should be different from zero. On Intel systems this function is implemented with the native intel fpatan instruction. If x is zero, an overflow error will occur. arsinh returns the inverse hyperbolic sine of its argument x. The arscsinh variant of this function is supplied for Delphi compatibility. None. artanh returns the inverse hyperbolic tangent of its argument x, where x should lie in the interval [-1,1], borders included. The arctanh variant of this function is supplied for Delphi compatibility. In case x is not in the interval [-1,1], an EInvalidArgument exception is raised. Ceil returns the lowest integer number greater than or equal to x. The absolute value of x should be less than maxint. If the asolute value of x is larger than maxint, an overflow error will occur. Cosh returns the hyperbolic cosine of it's argument {x}. None. Cotan returns the cotangent of it's argument x. x should be different from zero. If x is zero then a overflow error will occur. Cycletorad transforms it's argument cycle (an angle expressed in cycles) to radians. (1 cycle is $2 \pi$ radians). None. , Degtograd transforms it's argument deg (an angle in degrees) to grads. (90 degrees is 100 grad.) None. , Degtorad converts it's argument deg (an angle in degrees) to radians. (pi radians is 180 degrees) None. , Floor returns the largest integer smaller than or equal to x. The absolute value of x should be less than maxint. If x is larger than maxint, an overflow will occur. Frexp returns the mantissa and exponent of it's argument x in mantissa and exponent. None Gradtodeg converts its argument grad (an angle in grads) to degrees. (100 grad is 90 degrees) None. , Gradtorad converts its argument grad (an angle in grads) to radians. (200 grad is pi degrees). None. , Hypot returns the hypotenuse of the triangle where the sides adjacent to the square angle have lengths x and y. The function uses Pythagoras' rule for this. None. Intpower returns base to the power exponent, where exponent is an integer value. If base is zero and the exponent is negative, then an overflow error will occur. Ldexp returns $2^p x$. None. Lnxp1 returns the natural logarithm of 1+X. The result is more precise for small values of x. x should be larger than -1. If $x\leq -1$ then an EInvalidArgument exception will be raised. Log10 returns the 10-base logarithm of X. If x is less than or equal to 0 an 'invalid fpu operation' error will occur. Log2 returns the 2-base logarithm of X. If x is less than or equal to 0 an 'invalid fpu operation' error will occur. Logn returns the n-base logarithm of X. If x is less than or equal to 0 an 'invalid fpu operation' error will occur. Max returns the maximum of Int1 and Int2. None. MaxIntValue returns the largest integer out of the Data array. This function is provided for Delphi compatibility, use the function instead. None. Maxvalue returns the largest value in the data array with integer or float values. The return value has the same type as the elements of the array. The third and fourth forms accept a pointer to an array of N integer or float values. None. Mean returns the average value of data. The second form accepts a pointer to an array of N values. None. meanandstddev calculates the mean and standard deviation of data and returns the result in mean and stddev, respectively. Stddev is zero if there is only one value. The second form accepts a pointer to an array of N values. None. min returns the smallest value of Int1 and Int2; None. MinIntvalue returns the smallest value in the Data array. This function is provided for Delphi compatibility, use minvalue instead. None Minvalue returns the smallest value in the data array with integer or float values. The return value has the same type as the elements of the array. The third and fourth forms accept a pointer to an array of N integer or float values. None. momentskewkurtosis calculates the 4 first moments of the distribution of valuesin data and returns them in m1,m2,m3 and m4, as well as the skew and kurtosis. None. Norm calculates the Euclidian norm of the array of data. This equals sqrt(sumofsquares(data)). The second form accepts a pointer to an array of N values. None. Popnstddev returns the square root of the population variance of the values in the Data array. It returns zero if there is only one value. The second form of this function accepts a pointer to an array of N values. None. , Popnvariance returns the square root of the population variance of the values in the Data array. It returns zero if there is only one value. The second form of this function accepts a pointer to an array of N values. None. , power raises base to the power power. This is equivalent to exp(power*ln(base)). Therefore base should be non-negative. None. Radtocycle converts its argument rad (an angle expressed in radians) to an angle in cycles. (1 cycle equals 2 pi radians) None. , Radtodeg converts its argument rad (an angle expressed in radians) to an angle in degrees. (180 degrees equals pi radians) None. , Radtodeg converts its argument rad (an angle expressed in radians) to an angle in grads. (200 grads equals pi radians) None. , randg returns a random number which - when produced in large quantities - has a Gaussian distribution with mean mean and standarddeviation stddev. None. Sincos calculates the sine and cosine of the angle theta, and returns the result in sinus and cosinus. On Intel hardware, This calculation will be faster than making 2 calls to clculatet he sine and cosine separately. None. . Sinh returns the hyperbolic sine of its argument x. Stddev returns the standard deviation of the values in Data. It returns zero if there is only one value. The second form of the function accepts a pointer to an array of N values. None. Sum returns the sum of the values in the data array. The second form of the function accepts a pointer to an array of N values. None. , Sumofsquares returns the sum of the squares of the values in the data array. The second form of the function accepts a pointer to an array of N values. None. , sumsandsquares calculates the sum of the values and the sum of the squares of the values in the data array and returns the results in sum and sumofsquares. The second form of the function accepts a pointer to an array of N values. None. , Tan returns the tangent of x. If x (normalized) is pi/2 or 3pi/2 then an overflow will occur. Tanh returns the hyperbolic tangent of x. None. TotalVariance returns the total variance of the values in the data array. It returns zero if there is only one value. The second form of the function accepts a pointer to an array of N values. None. Variance returns the variance of the values in the data array. It returns zero if there is only one value. The second form of the function accepts a pointer to an array of N values. None.