Package Modelica.​Math.​Special
Library of special mathematical functions

Information

This sublibrary contains functions to compute often used mathematical operators that cannot be expressed analytically.

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

Package Contents

NameDescription
erfError function erf(u) = 2/sqrt(pi)*Integral_0_u exp(-t^2)*d
erfcComplementary error function erfc(u) = 1 - erf(u)
erfcInvInverse complementary error function: u = erfc(erfcInv(u))
erfInvInverse error function: u = erf(erfInv(u))
Internal …Internal utility functions that should not be directly utilized by the user
sincUnnormalized sinc function: sinc(u) = sin(u)/u

Function Modelica.​Math.​Special.​erf
Error function erf(u) = 2/sqrt(pi)*Integral_0_u exp(-t^2)*d

Information

Syntax

Special.erf(u);

Description

This function computes the error function erf(u) = 2/sqrt(pi)*Integral_0_u exp(-t^2)*dt numerically with a relative precision of about 1e-15. The implementation utilizes the formulation of the Boost library (53-bit implementation of erf.hpp, developed by John Maddock). Plot of the function:

For more details, see Wikipedia.

Example

  erf(0)    // = 0
  erf(10)   // = 1
  erf(0.5)  // = 0.520499877813047

See also

erfc, erfInv, erfcInv.

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

Inputs

TypeNameDescription
RealuInput argument

Outputs

TypeNameDescription
Realy= 2/sqrt(pi)*Integral_0_u exp(-t^2)*dt

Function Modelica.​Math.​Special.​erfc
Complementary error function erfc(u) = 1 - erf(u)

Information

Syntax

Special.erfc(u);

Description

This function computes the complementary error function erfc(u) = 1 - erf(u) with a relative precision of about 1e-15. The implementation utilizes the formulation of the Boost library (53-bit implementation of erf.hpp developed by John Maddock). Plot of the function:

If u is large and erf(u) is subtracted from 1.0, the result is not accurate. It is then better to use erfc(u). For more details, see Wikipedia.

Example

  erfc(0)    // = 1
  erfc(10)   // = 0
  erfc(0.5)  // = 0.4795001221869534

See also

erf, erfInv, erfcInv.

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

Inputs

TypeNameDescription
RealuInput argument

Outputs

TypeNameDescription
Realy= 1 - erf(u)

Function Modelica.​Math.​Special.​erfInv
Inverse error function: u = erf(erfInv(u))

Information

Syntax

Special.erfInv(u);

Description

This function computes the inverse of the error function erf(u) = 2/sqrt(pi)*Integral_0_u exp(-t^2)*dt numerically with a relative precision of about 1e-15. Therefore, u = erf(erfInv(u)). Input argument u must be in the range (otherwise an assertion is raised):

-1 ≤ u ≤ 1

If u = 1, the function returns Modelica.Constants.inf.
If u = -1, the function returns -Modelica.Constants.inf
The implementation utilizes the formulation of the Boost library (erf_inv.hpp, developed by John Maddock).
Plot of the function:

For more details, see Wikipedia.

Example

  erfInv(0)            // = 0
  erfInv(0.5)          // = 0.4769362762044699
  erfInv(0.999999)     // = 3.458910737275499
  erfInv(0.9999999999) // = 4.572824958544925

See also

erf, erfc, erfcInv.

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

Inputs

TypeNameDescription
RealuInput argument in the range -1 <= u <= 1

Outputs

TypeNameDescription
Realy= inverse of error function

Function Modelica.​Math.​Special.​erfcInv
Inverse complementary error function: u = erfc(erfcInv(u))

Information

Syntax

Special.erfInv(u);

Description

This function computes the inverse of the complementary error function erfc(u) = 1 - erf(u) with a relative precision of about 1e-15. Therefore, u = erfc(erfcInv(u)) and erfcInv(u) = erfInv(1 - u). Input argument u must be in the range (otherwise an assertion is raised):

0 ≤ u ≤ 2

If u = 2, the function returns -Modelica.Constants.inf.
If u = 0, the function returns Modelica.Constants.inf
The implementation utilizes the formulation of the Boost library (erf_inv.hpp, developed by John Maddock).
Plot of the function:

For more details, see Wikipedia.

Example

  erfcInv(1)         // = 0
  erfcInv(0.5)       // = 0.4769362762044699
  erfInv(1.999999)   // = -3.4589107372909473

See also

erf, erfc, erfInv.

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

Inputs

TypeNameDescription
RealuInput argument

Outputs

TypeNameDescription
RealyerfcInv(u)

Function Modelica.​Math.​Special.​sinc
Unnormalized sinc function: sinc(u) = sin(u)/u

Information

Syntax

Special.sinc(u);

Description

This function computes the unnormalized sinc function sinc(u) = sin(u)/u. The implementation utilizes a Taylor series approximation for small values of u. Plot of the function:

For more details, see Wikipedia.

Example

  sinc(0)   // = 1
  sinc(3)   // = 0.0470400026866224

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

Inputs

TypeNameDescription
RealuInput argument

Outputs

TypeNameDescription
Realy= sinc(u) = sin(u)/u

Generated 2018-12-12 12:14:35 EST by MapleSim.