Mathematical Functions

LINGO offers a number of standard, mathematical functions. These functions return a single result based on one or more scalar arguments. These functions are listed below.

Note: Not all mathematical functions are defined over all argument values. For example, @LOG( X - 5) is not defined for values of X less-than-or-equal-to 5.  In such cases, it may be helpful to bound variables so that all functions are defined, otherwise the solver may return a numeric error. For our @LOG example here, we would want to add a bound statement like @BND( 5.001, X, 100), to force X to lie within the range 5.001 and 100, forcing the @LOG function to be defined over the entire range for X.

@ABS( X)

Returns the absolute value of X.

@ACOS( X)

Returns the inverse cosine, or arccosine, of X, where X is an angle in radians.

@ACOSH( X)

Returns the inverse hyperbolic cosine of X, where X is an angle in radians.

@ASIN( X)

Returns the inverse sine, or arcsine, of X, where X is an angle in radians.

@ASINH( X)

Returns the inverse hyperbolic sine of X, where X is an angle in radians.

@ATAN( X)

Returns the inverse tangent, or arctangent, of X, where X is an angle in radians.

@ATAN2( X, Y)

Returns the inverse tangent of Y/X.

@ATANH( X)

Returns the inverse hyperbolic tangent of X, where X is an angle in radians.

@COS( X)

Returns the cosine of X, where X is an angle in radians.

@COSH( X)

Returns the hyperbolic cosine of X, where X is an angle in radians.

@EXP( X)

Returns e (2.718281...) raised to the power X.

@FLOOR( X)

This returns the integer part of X. To be specific, if X ≥ 0, @FLOOR returns the largest integer, I, such that I ≤ X. If X is negative, @FLOOR returns the most negative integer, I, such that I ≥ X.

@INT( X)

This returns the integer part of X. To be specific, if X ≥ 0, @INT returns the largest integer, I, such that I ≤ X. If X is negative, @INT returns the largest negative integer, I, such that X ≥ I.

@INTEGRAL( PROCEDURE, X, XL, XU, Y)

This function performs numeric integration using a variant of Simpson's Rule.  Argument PROCEDURE is the name of a procedure which calculates the function to be integrated.  The function's value should be placed in Y in the specified procedure.  The integral will be taken with respect to X over the interval [XL,XU].  At present, @INTEGRAL may only be used in calc sections.

Example 1:

In this example, we illustrate how to compute the integral of the standard normal distribution over the interval [-5,1.645].

MODEL:

 

! Takes the integral of the standard normal

distribution over the interval [-5,1.645];

 

PROCEDURE NORMAL:

  Y = @EXP( -X * X / 2) / (( 2 * @PI()) ^ 0.5);

ENDPROCEDURE

 

CALC:

  PROB = @INTEGRAL( NORMAL, X, -5, 1.645, Y);

ENDCALC

 

END

@LGM( X)

Returns the natural (base e) logarithm of the gamma function of X (i.e., log of (X - 1)!). It is extended to non-integer values of X by linear interpolation.

@LOG( X)

Returns the natural logarithm of X.

@LOG10( X)

Returns the base-10 logarithm of X.

@LOGB( X, B)

Returns the base B logarithm of X, e.g., @LOGB( 8, 2) = 3.

@MOD( X, Y)

Returns the value of X modulo Y, or, in other words, the remainder of an integer divide of X by Y.

@PI()

Returns the value of PI, i.e., 3.14159265....

@POW( X, Y)

Returns the value of X raised to the Y power.

@ROUND( X, N)

The @ROUND function rounds X to the closest number to X having N digits.  For example, if X is equal to 2.576 and N is 2, then @ROUND( X, N) will return 2.58.   If X is equal to -2.576 and N is 2, then @ROUND( X, N) will return -2.58.

@ROUNDDOWN( X, N)

The @ROUNDDOWN function rounds X down (towards 0) to the closest number to X  having N digits.  For example, if X is equal to 2.576 and N is 2, then @ROUNDDOWN( X, N) will return 2.57.   If X is equal to -2.576 and N is 2, then @ROUNDDOWN( X, N) will return -2.57.

@ROUNDUP( X, N)

The @ROUNDUP function rounds X up (away from 0) to the closest number to X  having N digits.  For example, if X is equal to 2.576 and N is 2, then @ROUNDUP( X, N) will return 2.58.   If X is equal to -2.576 and N is 2, then @ROUNDUP( X, N) will return -2.58.

@SIGN( X)

Returns -1 if X < 0, 0 if X = 0 and +1 if X > 0.

@SIN( X)

Returns the sine of X, where X is the angle in radians.

@SINH( X)

Returns the hyperbolic sine of X, where X is an angle in radians.

@SMAX( X1, X2,..., XN)

Returns the maximum value of X1, X2, ..., and XN.

@SMIN( X1, X2,..., XN)

Returns the minimum value of X1, X2, ..., and XN.

@SQR( X)

Returns the value of X squared.

@SQRT( X)

Returns the square root of X.

@TAN( X)

Returns the tangent of X, where X is the angle in radians.

@TANH( X)

Returns the hyperbolic tangent of X, where X is an angle in radians.