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 © 1998-2010, 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 http://www.modelica.org/licenses/ModelicaLicense2.

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

Package Content

Name	Description
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
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	Basic icon for mathematical function with y-axis on left side
baseIcon2	Basic icon for mathematical function with y-axis in middle
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

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

Inputs

Type	Name	Default	Description
Real	s1		First scalar
Real	s2		Second scalar
Real	eps	0	The two scalars are identical if abs(s1-s2) <= eps

Outputs

Type	Name	Description
Boolean	result	= true, if scalars are identical

Modelica definition

function isEqual 
  "Determine if two Real scalars are numerically identical"
  extends Modelica.Icons.Function;
  input Real s1 "First scalar";
  input Real s2 "Second scalar";
  input Real eps(min=0) = 0 
    "The two scalars are identical if abs(s1-s2) <= eps";
  output Boolean result "= true, if scalars are identical";
algorithm 
  result :=abs(s1 - s2) <= eps;
end isEqual;

Modelica.Math.sin

Information

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

Extends from baseIcon1 (Basic icon for mathematical function with y-axis on left side).

Inputs

Type	Name	Default	Description
Angle	u		[rad]

Outputs

Type	Name	Description
Real	y

Modelica definition

function sin "Sine"
  extends baseIcon1;
  input Modelica.SIunits.Angle u;
  output Real y;

external "builtin" y = sin(u);
end sin;

Modelica.Math.cos

Information

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

Extends from baseIcon1 (Basic icon for mathematical function with y-axis on left side).

Inputs

Type	Name	Default	Description
Angle	u		[rad]

Outputs

Type	Name	Description
Real	y

Modelica definition

function cos "Cosine"
  extends baseIcon1;
  input SI.Angle u;
  output Real y;

external "builtin" y = cos(u);
end cos;

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).

Extends from baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Angle	u		[rad]

Outputs

Type	Name	Description
Real	y

Modelica definition

function tan "Tangent (u shall not be -pi/2, pi/2, 3*pi/2, ...)"
  extends baseIcon2;
  input SI.Angle u;
  output Real y;

external "builtin" y = tan(u);
end tan;

Modelica.Math.asin

Information

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

Extends from baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Angle	y	[rad]

Modelica definition

function asin "Inverse sine (-1 <= u <= 1)"
  extends baseIcon2;
  input Real u;
  output SI.Angle y;

external "builtin" y = asin(u);
end asin;

Modelica.Math.acos

Information

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

Extends from baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Angle	y	[rad]

Modelica definition

function acos "Inverse cosine (-1 <= u <= 1)"
  extends baseIcon2;
  input Real u;
  output SI.Angle y;

external "builtin" y = acos(u);
end acos;

Modelica.Math.atan

Information

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

Extends from baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Angle	y	[rad]

Modelica definition

function atan "Inverse tangent"
  extends baseIcon2;
  input Real u;
  output SI.Angle y;

external "builtin" y = atan(u);
end atan;

Modelica.Math.atan2

Information

Extends from baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u1
Real	u2

Outputs

Type	Name	Description
Angle	y	[rad]

Modelica definition

function atan2 "Four quadrant inverse tangent"
  extends baseIcon2;
  input Real u1;
  input Real u2;
  output SI.Angle y;

external "builtin" y = atan2(u1, u2);
end atan2;

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:

Extends from Modelica.Math.baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u1
Real	u2
Angle	y0	0	y shall be in the range: -pi < y-y0 < pi [rad]

Outputs

Type	Name	Description
Angle	y	[rad]

Modelica definition

function atan3 
  "Four quadrant inverse tangent (select solution that is closest to given angle y0)"
  import Modelica.Math;
  extends Modelica.Math.baseIcon2;
  input Real u1;
  input Real u2;
  input Modelica.SIunits.Angle y0=0 "y shall be in the range: -pi < y-y0 < pi";
  output Modelica.SIunits.Angle y;

protected 
  Real pi = Modelica.Constants.pi;
  Real w;
algorithm 
  w :=Math.atan2(u1, u2);
  y := w + 2*pi*div(abs(w-y0)+pi,2*pi)*(if y0 > w then +1 else -1);
end atan3;

Modelica.Math.sinh

Information

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

Extends from baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Real	y

Modelica definition

function sinh "Hyperbolic sine"
  extends baseIcon2;
  input Real u;
  output Real y;

external "builtin" y = sinh(u);
end sinh;

Modelica.Math.cosh

Information

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

Extends from baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Real	y

Modelica definition

function cosh "Hyperbolic cosine"
  extends baseIcon2;
  input Real u;
  output Real y;

external "builtin" y = cosh(u);
end cosh;

Modelica.Math.tanh

Information

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

Extends from baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Real	y

Modelica definition

function tanh "Hyperbolic tangent"
  extends baseIcon2;
  input Real u;
  output Real y;

external "builtin" y = tanh(u);
end tanh;

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 < ∞):

Extends from Modelica.Math.baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Real	y

Modelica definition

function asinh "Inverse of sinh (area hyperbolic sine)"
  extends Modelica.Math.baseIcon2;
  input Real u;
  output Real y;

algorithm 
  y :=Modelica.Math.log(u + sqrt(u*u + 1));
end asinh;

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:

Extends from Modelica.Math.baseIcon1 (Basic icon for mathematical function with y-axis on left side).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Real	y

Modelica definition

function acosh "Inverse of cosh (area hyperbolic cosine)"
  import Modelica.Utilities.Streams.*;
  extends Modelica.Math.baseIcon1;
  input Real u;
  output Real y;

algorithm 
  assert(u>=1.0, "Input argument u (= " + String(u) + ") of acosh(u) must be >= 1.0");
  y :=Modelica.Math.log(u + sqrt(u*u - 1));
end acosh;

Modelica.Math.exp

Information

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

Extends from baseIcon2 (Basic icon for mathematical function with y-axis in middle).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Real	y

Modelica definition

function exp "Exponential, base e"
  extends baseIcon2;
  input Real u;
  output Real y;

external "builtin" y = exp(u);
end exp;

Modelica.Math.log

Information

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

Extends from baseIcon1 (Basic icon for mathematical function with y-axis on left side).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Real	y

Modelica definition

function log "Natural (base e) logarithm (u shall be > 0)"
  extends baseIcon1;
  input Real u;
  output Real y;

external "builtin" y = log(u);
end log;

Modelica.Math.log10

Information

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

Extends from baseIcon1 (Basic icon for mathematical function with y-axis on left side).

Inputs

Type	Name	Default	Description
Real	u

Outputs

Type	Name	Description
Real	y

Modelica definition

function log10 "Base 10 logarithm (u shall be > 0)"
  extends baseIcon1;
  input Real u;
  output Real y;

external "builtin" y = log10(u);
end log10;

Modelica.Math.baseIcon1

Information

Icon for a mathematical function, consisting of an y-axis on the left side. It is expected, that an x-axis is added and a plot of the function.

Modelica definition

partial function baseIcon1 
  "Basic icon for mathematical function with y-axis on left side"

end baseIcon1;

Modelica.Math.baseIcon2

Information

Icon for a mathematical function, consisting of an y-axis in the middle. It is expected, that an x-axis is added and a plot of the function.

Modelica definition

partial function baseIcon2 
  "Basic icon for mathematical function with y-axis in middle"

end baseIcon2;

Modelica.Math.tempInterpol1

Information

Inputs

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

Outputs

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

Modelica definition

function tempInterpol1 
  "Temporary function for linear interpolation (will be removed)"
  input Real u "input value (first column of table)";
  input Real table[:, :] "table to be interpolated";
  input Integer icol "column of table to be interpolated";
  output Real y "interpolated input value (icol column of table)";
protected 
  Integer i;
  Integer n "number of rows of table";
  Real u1;
  Real u2;
  Real y1;
  Real y2;
algorithm 
  n := size(table, 1);

  if n <= 1 then
    y := table[1, icol];

  else
    // Search interval

    if u <= table[1, 1] then
      i := 1;

    else
      i := 2;
      // Supports duplicate table[i, 1] values
      // in the interior to allow discontinuities.
      // Interior means that
      // if table[i, 1] = table[i+1, 1] we require i>1 and i+1<n

      while i < n and u >= table[i, 1] loop
        i := i + 1;

      end while;
      i := i - 1;

    end if;

    // Get interpolation data
    u1 := table[i, 1];
    u2 := table[i + 1, 1];
    y1 := table[i, icol];
    y2 := table[i + 1, icol];

    assert(u2 > u1, "Table index must be increasing");
    // Interpolate
    y := y1 + (y2 - y1)*(u - u1)/(u2 - u1);

  end if;

end tempInterpol1;

Modelica.Math.tempInterpol2

Information

Inputs

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

Outputs

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

Modelica definition

function tempInterpol2 
  "Temporary function for vectorized linear interpolation (will be removed)"

  input Real u "input value (first column of table)";
  input Real table[:, :] "table to be interpolated";
  input Integer icol[:] "column(s) of table to be interpolated";
  output Real y[1, size(icol, 1)] 
    "interpolated input value(s) (column(s) icol of table)";
protected 
  Integer i;
  Integer n "number of rows of table";
  Real u1;
  Real u2;
  Real y1[1, size(icol, 1)];
  Real y2[1, size(icol, 1)];
algorithm 
  n := size(table, 1);

  if n <= 1 then
    y := transpose([table[1, icol]]);

  else
    // Search interval

    if u <= table[1, 1] then
      i := 1;

    else
      i := 2;
      // Supports duplicate table[i, 1] values
      // in the interior to allow discontinuities.
      // Interior means that
      // if table[i, 1] = table[i+1, 1] we require i>1 and i+1<n

      while i < n and u >= table[i, 1] loop
        i := i + 1;

      end while;
      i := i - 1;

    end if;

    // Get interpolation data
    u1 := table[i, 1];
    u2 := table[i + 1, 1];
    y1 := transpose([table[i, icol]]);
    y2 := transpose([table[i + 1, icol]]);

    assert(u2 > u1, "Table index must be increasing");
    // Interpolate
    y := y1 + (y2 - y1)*(u - u1)/(u2 - u1);

  end if;

end tempInterpol2;