Modelica.Math

Library of mathematical functions (e.g., sin, cos) and of functions operating on vectors and matrices

Information

This package contains basic mathematical functions (such as sin(..)), as well as functions operating on vectors, matrices, nonlinear functions, and Boolean vectors.

Main Authors

Martin Otter and Marcus Baur
Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR)
Institut für Systemdynamik und Regelungstechnik (DLR-SR)
Forschungszentrum Oberpfaffenhofen
D-82234 Wessling
Germany
email: Martin.Otter@dlr.de

Extends from Modelica.Icons.Package (Icon for standard packages).

Package Content

Name	Description
$Modelica.Math.Vectors$ Vectors	Library of functions operating on vectors
$Modelica.Math.BooleanVectors$ BooleanVectors	Library of functions operating on Boolean vectors
$Modelica.Math.Matrices$ Matrices	Library of functions operating on matrices
$Modelica.Math.Nonlinear$ Nonlinear	Library of functions operating on nonlinear equations
$Modelica.Math.Polynomials$ Polynomials	Library of functions operating on polynomials (including polynomial fitting)
$Modelica.Math.Random$ Random	Library of functions for generating random numbers
$Modelica.Math.Distributions$ Distributions	Library of distribution functions
$Modelica.Math.Special$ Special	Library of special mathematical functions
$Modelica.Math.FastFourierTransform$ FastFourierTransform	Library of functions for the Fast Fourier Transform (FFT)
$Modelica.Math.Icons$ Icons	Icons for Math
$Modelica.Math.isEqual$ isEqual	Determine if two Real scalars are numerically identical
$Modelica.Math.isPowerOf2$ isPowerOf2	Determine if the integer input is a power of 2
$Modelica.Math.sin$ sin	Sine
$Modelica.Math.cos$ cos	Cosine
$Modelica.Math.tan$ tan	Tangent (u shall not be -pi/2, pi/2, 3*pi/2, ...)
$Modelica.Math.asin$ asin	Inverse sine (-1 <= u <= 1)
$Modelica.Math.acos$ acos	Inverse cosine (-1 <= u <= 1)
$Modelica.Math.atan$ atan	Inverse tangent
$Modelica.Math.atan2$ atan2	Four quadrant inverse tangent
$Modelica.Math.atan3$ atan3	Four quadrant inverse tangent (select solution that is closest to given angle y0)
$Modelica.Math.sinh$ sinh	Hyperbolic sine
$Modelica.Math.cosh$ cosh	Hyperbolic cosine
$Modelica.Math.tanh$ tanh	Hyperbolic tangent
$Modelica.Math.asinh$ asinh	Inverse of sinh (area hyperbolic sine)
$Modelica.Math.acosh$ acosh	Inverse of cosh (area hyperbolic cosine)
$Modelica.Math.exp$ exp	Exponential, base e
$Modelica.Math.log$ log	Natural (base e) logarithm (u shall be > 0)
$Modelica.Math.log10$ log10	Base 10 logarithm (u shall be > 0)
$Modelica.Math.wrapAngle$ wrapAngle	Wrap angle to interval ]-pi,pi] or [0,2*pi[

$Modelica.Math.isEqual$ Modelica.Math.isEqual

Determine if two Real scalars are numerically identical

Information

Syntax

Math.isEqual(s1, s2);
Math.isEqual(s1, s2, eps=0);

Description

The function call "Math.isEqual(s1, s2)" returns true, if the two Real scalars s1 and s2 are identical. Otherwise the function returns false. The equality check is performed by "abs(s1-s2) ≤ eps", where "eps" can be provided as third argument of the function. Default is "eps = 0".

Example

  Real s1 = 2.0;
  Real s2 = 2.0;
  Real s3 = 2.000001;
  Boolean result;
algorithm
  result := Math.isEqual(s1,s2);     // = true
  result := Math.isEqual(s1,s3);     // = false
  result := Math.isEqual(s1,s3,0.1); // = true

Inputs

Name	Description
s1	First scalar
s2	Second scalar
eps	The two scalars are identical if abs(s1-s2) <= eps

Outputs

Name	Description
result	= true, if scalars are identical

$Modelica.Math.isPowerOf2$ Modelica.Math.isPowerOf2

Determine if the integer input is a power of 2

Information

Syntax

Math.isPowerOf2(i);

Description

The function call "Math.isPowerOf2(i)" returns true, if the Integer input i is a power of 2. Otherwise the function returns false. The Integer input has to be >=1.

Example

  Integer i1 = 1;
  Integer i2 = 4;
  Integer i3 = 9;
  Boolean result;
algorithm
  result := Math.isPowerOf2(i1);     // = true 2^0
  result := Math.isPowerOf2(i2);     // = true 2^2
  result := Math.isPowerOf2(i3);     // = false

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Name	Description
i	Integer scalar

Outputs

Name	Description
result	= true, if integer scalar is a power of 2

$Modelica.Math.sin$ Modelica.Math.sin

Sine

Information

This function returns y = sin(u), with -∞ < u < ∞:

Extends from Modelica.Math.Icons.AxisLeft (Basic icon for mathematical function with y-axis on left side).

Inputs

Name	Description
u	Independent variable [rad]

Outputs

Name	Description
y	Dependent variable y=sin(u)

$Modelica.Math.cos$ Modelica.Math.cos

Cosine

Information

This function returns y = cos(u), with -∞ < u < ∞:

Extends from Modelica.Math.Icons.AxisLeft (Basic icon for mathematical function with y-axis on left side).

Inputs

Name	Description
u	Independent variable [rad]

Outputs

Name	Description
y	Dependent variable y=cos(u)

$Modelica.Math.tan$ Modelica.Math.tan

Tangent (u shall not be -pi/2, pi/2, 3*pi/2, ...)

Information

This function returns y = tan(u), with -∞ < u < ∞ (if u is a multiple of (2n-1)*pi/2, y = tan(u) is +/- infinity).

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Independent variable [rad]

Outputs

Name	Description
y	Dependent variable y=tan(u)

$Modelica.Math.asin$ Modelica.Math.asin

Inverse sine (-1 <= u <= 1)

Information

This function returns y = asin(u), with -1 ≤ u ≤ +1:

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=asin(u) [rad]

$Modelica.Math.acos$ Modelica.Math.acos

Inverse cosine (-1 <= u <= 1)

Information

This function returns y = acos(u), with -1 ≤ u ≤ +1:

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=acos(u) [rad]

$Modelica.Math.atan$ Modelica.Math.atan

Inverse tangent

Information

This function returns y = atan(u), with -∞ < u < ∞:

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=atan(u) [rad]

$Modelica.Math.atan2$ Modelica.Math.atan2

Four quadrant inverse tangent

Information

This function returns y = atan2(u1,u2) such that tan(y) = u1/u2 and y is in the range -pi < y ≤ pi. u2 may be zero, provided u1 is not zero. Usually u1, u2 is provided in such a form that u1 = sin(y) and u2 = cos(y):

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u1	First independent variable
u2	Second independent variable

Outputs

Name	Description
y	Dependent variable y=atan2(u1, u2)=atan(u1/u2) [rad]

$Modelica.Math.atan3$ Modelica.Math.atan3

Four quadrant inverse tangent (select solution that is closest to given angle y0)

Information

This function returns y = atan3(u1,u2,y0) such that tan(y) = u1/u2 and y is in the range: -pi < y-y0 ≤ pi.
u2 may be zero, provided u1 is not zero. The difference to Modelica.Math.atan2(..) is the optional third argument y0 that allows to specify which of the infinite many solutions shall be returned:

Note, for the default case (y0=0), exactly the same result as with atan2(..) is returned.

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u1	First independent variable
u2	Second independent variable
y0	y shall be in the range: -pi < y-y0 <= pi [rad]

Outputs

Name	Description
y	Dependent variable y=atan3(u1, u2, y0)=atan(u1/u2) [rad]

$Modelica.Math.sinh$ Modelica.Math.sinh

Hyperbolic sine

Information

This function returns y = sinh(u), with -∞ < u < ∞:

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=sinh(u)

$Modelica.Math.cosh$ Modelica.Math.cosh

Hyperbolic cosine

Information

This function returns y = cosh(u), with -∞ < u < ∞:

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=cosh(u)

$Modelica.Math.tanh$ Modelica.Math.tanh

Hyperbolic tangent

Information

This function returns y = tanh(u), with -∞ < u < ∞:

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=tanh(u)

$Modelica.Math.asinh$ Modelica.Math.asinh

Inverse of sinh (area hyperbolic sine)

Information

The function returns the area hyperbolic sine of its input argument u. This inverse of sinh(..) is unique and there is no restriction on the input argument u of asinh(u) (-∞ < u < ∞):

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=asinh(u)

$Modelica.Math.acosh$ Modelica.Math.acosh

Inverse of cosh (area hyperbolic cosine)

Information

This function returns the area hyperbolic cosine of its input argument u. The valid range of u is

+1 ≤ u < +∞

If the function is called with u < 1, an error occurs. The function cosh(u) has two inverse functions (the curve looks similar to a sqrt(..) function). acosh(..) returns the inverse that is positive. At u=1, the derivative dy/du is infinite. Therefore, this function should not be used in a model, if u can become close to 1:

Extends from Modelica.Math.Icons.AxisLeft (Basic icon for mathematical function with y-axis on left side).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=acosh(u)

$Modelica.Math.exp$ Modelica.Math.exp

Exponential, base e

Information

This function returns y = exp(u), with -∞ < u < ∞:

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=exp(u)

$Modelica.Math.log$ Modelica.Math.log

Natural (base e) logarithm (u shall be > 0)

Information

This function returns y = log(10) (the natural logarithm of u), with u > 0:

Extends from Modelica.Math.Icons.AxisLeft (Basic icon for mathematical function with y-axis on left side).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=ln(u)

$Modelica.Math.log10$ Modelica.Math.log10

Base 10 logarithm (u shall be > 0)

Information

This function returns y = log10(u), with u > 0:

Extends from Modelica.Math.Icons.AxisLeft (Basic icon for mathematical function with y-axis on left side).

Inputs

Name	Description
u	Independent variable

Outputs

Name	Description
y	Dependent variable y=lg(u)

$Modelica.Math.wrapAngle$ Modelica.Math.wrapAngle

Wrap angle to interval ]-pi,pi] or [0,2*pi[

Information

This function wraps the input angle u to the interval ]-pi,pi], if positiveRange == false. Otherwise the input angle u is wrapped to the interval [0,2*pi[.

Extends from Modelica.Math.Icons.AxisCenter (Basic icon for mathematical function with y-axis in the center).

Inputs

Name	Description
u	Input angle [rad]
positiveRange	Use only positive output range, if true

Outputs

Name	Description
y	Wrapped output angle [rad]

Automatically generated Thu Oct 1 16:08:15 2020.

Modelica.Math

Information

Main Authors

Package Content

Modelica.Math.isEqual

Information

Syntax

Description

Example

See also

Inputs

Outputs

Modelica.Math.isPowerOf2

Information

Syntax

Description

Example

Inputs

Outputs

Modelica.Math.sin

Information

Inputs

Outputs

Modelica.Math.cos

Information

Inputs

Outputs

Modelica.Math.tan

Information

Inputs

Outputs

Modelica.Math.asin

Information

Inputs

Outputs

Modelica.Math.acos

Information

Inputs

Outputs

Modelica.Math.atan

Information

Inputs

Outputs

Modelica.Math.atan2

Information

Inputs

Outputs

Modelica.Math.atan3

Information

Inputs

Outputs

Modelica.Math.sinh

Information

Inputs

Outputs

Modelica.Math.cosh

Information

Inputs

Outputs

Modelica.Math.tanh

Information

Inputs

Outputs

Modelica.Math.asinh

Information

Inputs

Outputs

Modelica.Math.acosh

Information

Inputs

Outputs

Modelica.Math.exp

Information

Inputs

Outputs

Modelica.Math.log

Information

Inputs

Outputs

Modelica.Math.log10

$Modelica.Math.isEqual$ Modelica.Math.isEqual

$Modelica.Math.isPowerOf2$ Modelica.Math.isPowerOf2

$Modelica.Math.sin$ Modelica.Math.sin

$Modelica.Math.cos$ Modelica.Math.cos

$Modelica.Math.tan$ Modelica.Math.tan

$Modelica.Math.asin$ Modelica.Math.asin

$Modelica.Math.acos$ Modelica.Math.acos

$Modelica.Math.atan$ Modelica.Math.atan

$Modelica.Math.atan2$ Modelica.Math.atan2

$Modelica.Math.atan3$ Modelica.Math.atan3

$Modelica.Math.sinh$ Modelica.Math.sinh

$Modelica.Math.cosh$ Modelica.Math.cosh

$Modelica.Math.tanh$ Modelica.Math.tanh

$Modelica.Math.asinh$ Modelica.Math.asinh

$Modelica.Math.acosh$ Modelica.Math.acosh

$Modelica.Math.exp$ Modelica.Math.exp

$Modelica.Math.log$ Modelica.Math.log

$Modelica.Math.log10$ Modelica.Math.log10

$Modelica.Math.wrapAngle$ Modelica.Math.wrapAngle