Modelica.Math.Polynomials

Library of functions operating on polynomials (including polynomial fitting)

Information

This package contains functions to operate on polynomials, in particular to determine the derivative and the integral of a polynomial and to use a polynomial to fit a given set of data points.

Extends from Modelica.Icons.FunctionsPackage (Icon for packages containing functions).

Package Content

Name	Description
$Modelica.Math.Polynomials.evaluate$ evaluate	Evaluate polynomial at a given abscissa value
$Modelica.Math.Polynomials.evaluateWithRange$ evaluateWithRange	Evaluate polynomial at a given abscissa value with linear extrapolation outside of the defined range
$Modelica.Math.Polynomials.derivative$ derivative	Derivative of polynomial
$Modelica.Math.Polynomials.derivativeValue$ derivativeValue	Value of derivative of polynomial at abscissa value u
$Modelica.Math.Polynomials.secondDerivativeValue$ secondDerivativeValue	Value of 2nd derivative of polynomial at abscissa value u
$Modelica.Math.Polynomials.integral$ integral	Indefinite integral of polynomial p(u)
$Modelica.Math.Polynomials.integralValue$ integralValue	Integral of polynomial p(u) from u_low to u_high
$Modelica.Math.Polynomials.fitting$ fitting	Computes the coefficients of a polynomial that fits a set of data points in a least-squares sense
$Modelica.Math.Polynomials.evaluate_der$ evaluate_der	Evaluate derivative of polynomial at a given abscissa value
$Modelica.Math.Polynomials.evaluateWithRange_der$ evaluateWithRange_der	Evaluate derivative of polynomial at a given abscissa value with extrapolation outside of the defined range
$Modelica.Math.Polynomials.integralValue_der$ integralValue_der	Time derivative of integral of polynomial p(u) from u_low to u_high, assuming only u_high as time-dependent (Leibniz rule)
$Modelica.Math.Polynomials.derivativeValue_der$ derivativeValue_der	Time derivative of derivative of polynomial
$Modelica.Math.Polynomials.roots$ roots	Compute zeros of a polynomial where the highest coefficient is assumed as not to be zero

$Modelica.Math.Polynomials.evaluate$ Modelica.Math.Polynomials.evaluate

Evaluate polynomial at a given abscissa value

Information

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

Inputs

Name	Description
p[:]	Polynomial coefficients (p[1] is coefficient of highest power)
u	Abscissa value

Outputs

Name	Description
y	Value of polynomial at u

$Modelica.Math.Polynomials.evaluateWithRange$ Modelica.Math.Polynomials.evaluateWithRange

Evaluate polynomial at a given abscissa value with linear extrapolation outside of the defined range

Information

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

Inputs

Name	Description
p[:]	Polynomial coefficients (p[1] is coefficient of highest power)
uMin	Polynomial valid in the range uMin .. uMax
uMax	Polynomial valid in the range uMin .. uMax
u	Abscissa value

Outputs

Name	Description
y	Value of polynomial at u. Outside of uMin,uMax, linear extrapolation is used

$Modelica.Math.Polynomials.derivative$ Modelica.Math.Polynomials.derivative

Derivative of polynomial

Information

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

Inputs

Name	Description
p1[:]	Polynomial coefficients (p1[1] is coefficient of highest power)

Outputs

Name	Description
p2[size(p1, 1) - 1]	Derivative of polynomial p1

$Modelica.Math.Polynomials.derivativeValue$ Modelica.Math.Polynomials.derivativeValue

Value of derivative of polynomial at abscissa value u

Information

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

Inputs

Name	Description
p[:]	Polynomial coefficients (p[1] is coefficient of highest power)
u	Abscissa value

Outputs

Name	Description
y	Value of derivative of polynomial at u

$Modelica.Math.Polynomials.secondDerivativeValue$ Modelica.Math.Polynomials.secondDerivativeValue

Value of 2nd derivative of polynomial at abscissa value u

Information

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

Inputs

Name	Description
p[:]	Polynomial coefficients (p[1] is coefficient of highest power)
u	Abscissa value

Outputs

Name	Description
y	Value of 2nd derivative of polynomial at u

$Modelica.Math.Polynomials.integral$ Modelica.Math.Polynomials.integral

Indefinite integral of polynomial p(u)

Information

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

Inputs

Name	Description
p1[:]	Polynomial coefficients (p1[1] is coefficient of highest power)

Outputs

Name	Description
p2[size(p1, 1) + 1]	Polynomial coefficients of indefinite integral of polynomial p1 (polynomial p2 + C is the indefinite integral of p1, where C is an arbitrary constant)

$Modelica.Math.Polynomials.integralValue$ Modelica.Math.Polynomials.integralValue

Integral of polynomial p(u) from u_low to u_high

Information

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

Inputs

Name	Description
p[:]	Polynomial coefficients
u_high	High integrand value
u_low	Low integrand value, default 0

Outputs

Name	Description
integral	Integral of polynomial p from u_low to u_high

$Modelica.Math.Polynomials.fitting$ Modelica.Math.Polynomials.fitting

Computes the coefficients of a polynomial that fits a set of data points in a least-squares sense

Information

Polynomials.fitting(u,y,n) computes the coefficients of a polynomial p(u) of degree "n" that fits the data "p(u[i]) - y[i]" in a least squares sense. The polynomial is returned as a vector p[n+1] that has the following definition:

p(u) = p[1]*u^n + p[2]*u^(n-1) + ... + p[n]*u + p[n+1];

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

Inputs

Name	Description
u[:]	Abscissa data values
y[size(u, 1)]	Ordinate data values
n	Order of desired polynomial that fits the data points (u,y)

Outputs

Name	Description
p[n + 1]	Polynomial coefficients of polynomial that fits the date points

$Modelica.Math.Polynomials.evaluate_der$ Modelica.Math.Polynomials.evaluate_der

Evaluate derivative of polynomial at a given abscissa value

Information

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

Inputs

Name	Description
p[:]	Polynomial coefficients (p[1] is coefficient of highest power)
u	Abscissa value
du	Delta of abscissa value

Outputs

Name	Description
dy	Value of derivative of polynomial at u

$Modelica.Math.Polynomials.evaluateWithRange_der$ Modelica.Math.Polynomials.evaluateWithRange_der

Evaluate derivative of polynomial at a given abscissa value with extrapolation outside of the defined range

Information

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

Inputs

Name	Description
p[:]	Polynomial coefficients (p[1] is coefficient of highest power)
uMin	Polynomial valid in the range uMin .. uMax
uMax	Polynomial valid in the range uMin .. uMax
u	Abscissa value
du	Delta of abscissa value

Outputs

Name	Description
dy	Value of derivative of polynomial at u

$Modelica.Math.Polynomials.integralValue_der$ Modelica.Math.Polynomials.integralValue_der

Time derivative of integral of polynomial p(u) from u_low to u_high, assuming only u_high as time-dependent (Leibniz rule)

Information

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

Inputs

Name	Description
p[:]	Polynomial coefficients
u_high	High integrand value
u_low	Low integrand value, default 0
du_high	High integrand value
du_low	Low integrand value, default 0

Outputs

Name	Description
dintegral	Integral of polynomial p from u_low to u_high

$Modelica.Math.Polynomials.derivativeValue_der$ Modelica.Math.Polynomials.derivativeValue_der

Time derivative of derivative of polynomial

Information

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

Inputs

Name	Description
p[:]	Polynomial coefficients (p[1] is coefficient of highest power)
u	Abscissa value
du	Delta of abscissa value

Outputs

Name	Description
dy	Time-derivative of derivative of polynomial w.r.t. input variable at u

$Modelica.Math.Polynomials.roots$ Modelica.Math.Polynomials.roots

Compute zeros of a polynomial where the highest coefficient is assumed as not to be zero

Information

Syntax

r = Polynomials.roots(p);

Description

This function computes the roots of a polynomial P of x

P = p[1]*x^n + p[2]*x^(n-1) + ... + p[n-1]*x + p[n+1];

with the coefficient vector p. It is assumed that the first element of p is not zero, i.e., that the polynomial is of order size(p,1)-1.

To compute the roots, the eigenvalues of the corresponding companion matrix C

       |-p[2]/p[1]  -p[3]/p[1]  ...  -p[n-2]/p[1]  -p[n-1]/p[1]  -p[n]/p[1] |
       |    1            0                0               0           0     |
       |    0            1      ...       0               0           0     |
C =    |    .            .      ...       .               .           .     |
       |    .            .      ...       .               .           .     |
       |    0            0      ...       0               1           0     |

are calculated. These are the roots of the polynomial.
Since the companion matrix has already Hessenberg form, the transformation to Hessenberg form has not to be performed. Function eigenvaluesHessenberg
provides efficient eigenvalue computation for those matrices.

Example

r = roots({1,2,3});
// r = [-1.0,  1.41421356237309;
//      -1.0, -1.41421356237309]
// which corresponds to the roots: -1.0 +/- j*1.41421356237309

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

Inputs

Name	Description
p[:]	Vector with polynomial coefficients p[1]x^n + p[2]x^(n-1) + p[n]*x +p[n-1]

Outputs

Name	Description
roots[max(0, size(p, 1) - 1), 2]	roots[:,1] and roots[:,2] are the real and imaginary parts of the roots of polynomial p

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