Modelica.Math

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 Robotik und Mechatronik
Postfach 1116
D-82230 Wessling
Germany
email: Martin.Otter@dlr.de

Copyright © 1998-2013, Modelica Association and DLR.

This Modelica package is free software and the use is completely at your own risk; it can be redistributed and/or modified under the terms of the Modelica License 2. For license conditions (including the disclaimer of warranty) see Modelica.UsersGuide.ModelicaLicense2 or visit https://www.modelica.org/licenses/ModelicaLicense2.

Package Content

Name	Description
Icons	Icons for Math
Vectors	Library of functions operating on vectors
Matrices	Library of functions operating on matrices
Nonlinear	Library of functions operating on nonlinear equations
BooleanVectors	Library of functions operating on Boolean vectors
isEqual	Determine if two Real scalars are numerically identical
isPowerOf2	Determine if the integer input is a power of 2
sin	Sine
cos	Cosine
tan	Tangent (u shall not be -pi/2, pi/2, 3*pi/2, ...)
asin	Inverse sine (-1 <= u <= 1)
acos	Inverse cosine (-1 <= u <= 1)
atan	Inverse tangent
atan2	Four quadrant inverse tangent
atan3	Four quadrant inverse tangent (select solution that is closest to given angle y0)
sinh	Hyperbolic sine
cosh	Hyperbolic cosine
tanh	Hyperbolic tangent
asinh	Inverse of sinh (area hyperbolic sine)
acosh	Inverse of cosh (area hyperbolic cosine)
exp	Exponential, base e
log	Natural (base e) logarithm (u shall be > 0)
log10	Base 10 logarithm (u shall be > 0)
baseIcon1	This icon will be removed in future Modelica versions, use Modelica.Math.Icons.AxisLeft instead.
baseIcon2	This icon will be removed in future Modelica versions, use Modelica.Math.Icons.AxisCenter instead.
tempInterpol1	Temporary function for linear interpolation (will be removed)
tempInterpol2	Temporary function for vectorized linear interpolation (will be removed)

Modelica.Math.isEqual

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

See also

Vectors.isEqual, Matrices.isEqual, Strings.isEqual

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

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

Inputs

Name	Description
i	Integer scalar

Outputs

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

Modelica.Math.sin

Information

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

Inputs

Name	Description
u	[rad]

Outputs

Name	Description
y

Modelica.Math.cos

Information

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

Inputs

Name	Description
u	[rad]

Outputs

Name	Description
y

Modelica.Math.tan

Information

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

Inputs

Name	Description
u	[rad]

Outputs

Name	Description
y

Modelica.Math.asin

Information

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

Inputs

Name	Description
u

Outputs

Name	Description
y	[rad]

Modelica.Math.acos

Information

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

Inputs

Name	Description
u

Outputs

Name	Description
y	[rad]

Modelica.Math.atan

Information

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

Inputs

Name	Description
u

Outputs

Name	Description
y	[rad]

Modelica.Math.atan2

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):

Inputs

Name	Description
u1
u2

Outputs

Name	Description
y	[rad]

Modelica.Math.atan3

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.

Inputs

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

Outputs

Name	Description
y	[rad]

Modelica.Math.sinh

Information

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

Inputs

Name	Description
u

Outputs

Name	Description
y

Modelica.Math.cosh

Information

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

Inputs

Name	Description
u

Outputs

Name	Description
y

Modelica.Math.tanh

Information

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

Inputs

Name	Description
u

Outputs

Name	Description
y

Modelica.Math.asinh

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 < ∞):

Inputs

Name	Description
u

Outputs

Name	Description
y

Modelica.Math.acosh

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:

Inputs

Name	Description
u

Outputs

Name	Description
y

Modelica.Math.exp

Information

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

Inputs

Name	Description
u

Outputs

Name	Description
y

Modelica.Math.log

Information

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

Inputs

Name	Description
u

Outputs

Name	Description
y

Modelica.Math.log10

Information

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

Inputs

Name	Description
u

Outputs

Name	Description
y

Modelica.Math.baseIcon1

Information

This icon will be removed in future versions of the Modelica Standard Library. Instead the icon Modelica.Math.Icons.AxisLeft shall be used.

Modelica.Math.baseIcon2

Information

This icon will be removed in future versions of the Modelica Standard Library. Instead the icon Modelica.Math.Icons.AxisCenter shall be used.

Modelica.Math.tempInterpol1

Information

Inputs

Name	Description
u	input value (first column of table)
table[:, :]	table to be interpolated
icol	column of table to be interpolated

Outputs

Name	Description
y	interpolated input value (icol column of table)

Modelica.Math.tempInterpol2

Information

Inputs

Name	Description
u	input value (first column of table)
table[:, :]	table to be interpolated
icol[:]	column(s) of table to be interpolated

Outputs

Name	Description
y[1, size(icol, 1)]	interpolated input value(s) (column(s) icol of table)