Package Modelica.​Math.​Vectors
Library of functions operating on vectors

Information

Library content

This library provides functions operating on vectors:

See also

Matrices

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

Package Contents

NameDescription
findFind element in a vector
interpolateInterpolate linearly in a vector
isEqualDetermine if two Real vectors are numerically identical
lengthReturn length of a vector (better as norm(), if further symbolic processing is performed)
normReturn the p-norm of a vector
normalizeReturn normalized vector such that length = 1 and prevent zero-division for zero vector
normalizeWithAssertReturn normalized vector such that length = 1 (trigger an assert for zero vector)
relNodePositionsReturn vector of relative node positions (0..1)
reverseReverse vector elements (e.g., v[1] becomes last element)
sortSort elements of vector in ascending or descending order
toStringConvert a real vector in to a string representation
Utilities …Utility functions that should not be directly utilized by the user

Function Modelica.​Math.​Vectors.​toString
Convert a real vector in to a string representation

Information

Syntax

Vectors.toString(v);
Vectors.toString(v,name="",significantDigits=6);

Description

The function call "Vectors.toString(v)" returns the string representation of vector v. With the optional arguments "name" and "significantDigits" a name and the number of the digits are defined. The default values of "name" and "significantDigits" are "" and 6 respectively. If name=="" (empty string) then the prefix "<name> =" is left out at the output-string.

Example

  v = {2.12, -4.34, -2.56, -1.67};
  toString(v);
                         // = "
                         //           2.12
                         //          -4.34
                         //          -2.56
                         //          -1.67"
  toString(v,"vv",1);
                         // = "vv =
                         //           2
                         //          -4
                         //          -3
                         //          -2"

See also

Matrices.toString,

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

Inputs

TypeNameDescription
Realv[:]Real vector
StringnameIndependent variable name used for printing
IntegersignificantDigitsNumber of significant digits that are shown

Outputs

TypeNameDescription
Strings 

Function Modelica.​Math.​Vectors.​isEqual
Determine if two Real vectors are numerically identical

Information

Syntax

Vectors.isEqual(v1, v2);
Vectors.isEqual(v1, v2, eps=0);

Description

The function call "Vectors.isEqual(v1, v2)" returns true, if the two Real vectors v1 and v2 have the same dimensions and the same elements. Otherwise the function returns false. Two elements e1 and e2 of the two vectors are checked on equality by the test "abs(e1-e2) ≤ eps", where "eps" can be provided as third argument of the function. Default is "eps = 0".

Example

  Real v1[3] = {1, 2, 3};
  Real v2[4] = {1, 2, 3, 4};
  Real v3[3] = {1, 2, 3.0001};
  Boolean result;
algorithm
  result := Vectors.isEqual(v1,v2);     // = false
  result := Vectors.isEqual(v1,v3);     // = false
  result := Vectors.isEqual(v1,v1);     // = true
  result := Vectors.isEqual(v1,v3,0.1); // = true

See also

Vectors.find, Matrices.isEqual, Strings.isEqual

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

Inputs

TypeNameDescription
Realv1[:]First vector
Realv2[:]Second vector (may have different length as v1)
RealepsTwo elements e1 and e2 of the two vectors are identical if abs(e1-e2) <= eps

Outputs

TypeNameDescription
Booleanresult= true, if vectors have the same length and the same elements

Function Modelica.​Math.​Vectors.​norm
Return the p-norm of a vector

Information

Syntax

Vectors.norm(v);
Vectors.norm(v,p=2);   // 1 ≤ p ≤ ∞

Description

The function call "Vectors.norm(v)" returns the Euclidean norm "sqrt(v*v)" of vector v. With the optional second argument "p", any other p-norm can be computed:

function Vectors.norm

Besides the Euclidean norm (p=2), also the 1-norm and the infinity-norm are sometimes used:

1-norm = sum(abs(v)) norm(v,1)
2-norm = sqrt(v*v) norm(v) or norm(v,2)
infinity-norm = max(abs(v)) norm(v,Modelica.Constants.inf)

Note, for any vector norm the following inequality holds: 

norm(v1+v2,p) ≤ norm(v1,p) + norm(v2,p)

Example

  v = {2, -4, -2, -1};
  norm(v,1);    // = 9
  norm(v,2);    // = 5
  norm(v);      // = 5
  norm(v,10.5); // = 4.00052597412635
  norm(v,Modelica.Constants.inf);  // = 4

See also

Matrices.norm

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

Inputs

TypeNameDescription
Realv[:]Real vector
RealpType of p-norm (often used: 1, 2, or Modelica.Constants.inf)

Outputs

TypeNameDescription
Realresultp-norm of vector v

Function Modelica.​Math.​Vectors.​length
Return length of a vector (better as norm(), if further symbolic processing is performed)

Information

Syntax

Vectors.length(v);

Description

The function call "Vectors.length(v)" returns the Euclidean length "sqrt(v*v)" of vector v. The function call is equivalent to Vectors.norm(v). The advantage of length(v) over norm(v)"is that function length(..) is implemented in one statement and therefore the function is usually automatically inlined. Further symbolic processing is therefore possible, which is not the case with function norm(..).

Example

  v = {2, -4, -2, -1};
  length(v);  // = 5

See also

Vectors.norm

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

Inputs

TypeNameDescription
Realv[:]Real vector

Outputs

TypeNameDescription
RealresultLength of vector v

Function Modelica.​Math.​Vectors.​normalize
Return normalized vector such that length = 1 and prevent zero-division for zero vector

Information

Syntax

Vectors.normalize(v);
Vectors.normalize(v,eps=100*Modelica.Constants.eps);

Description

The function call "Vectors.normalize(v)" returns the unit vector "v/length(v)" of vector v. If length(v) is close to zero (more precisely, if length(v) < eps), v/eps is returned in order to avoid a division by zero. For many applications this is useful, because often the unit vector e = v/length(v) is used to compute a vector x*e, where the scalar x is in the order of length(v), i.e., x*e is small, when length(v) is small and then it is fine to replace e by v to avoid a division by zero.

Since the function has the "Inline" annotation, it is usually inlined and symbolic processing is applied.

Example

  normalize({1,2,3});  // = {0.267, 0.534, 0.802}
  normalize({0,0,0});  // = {0,0,0}

See also

Vectors.length, Vectors.normalizeWithAssert

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

Inputs

TypeNameDescription
Realv[:]Real vector
Realepsif |v| < eps then result = v/eps

Outputs

TypeNameDescription
Realresult[size(v, 1)]Input vector v normalized to length=1

Function Modelica.​Math.​Vectors.​normalizeWithAssert
Return normalized vector such that length = 1 (trigger an assert for zero vector)

Information

Syntax

Vectors.normalizeWithAssert(v);

Description

The function call "Vectors.normalizeWithAssert(v)" returns the unit vector "v/sqrt(v*v)" of vector v. If vector v is a zero vector, an assert is triggered.

Since the function has the "Inline" annotation, it is usually inlined and symbolic processing is applied.

Example

  normalizeWithAssert({1,2,3});  // = {0.267, 0.534, 0.802}
  normalizeWithAssert({0,0,0});  // error (an assert is triggered)

See also

Vectors.length, Vectors.normalize

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

Inputs

TypeNameDescription
Realv[:]Real vector

Outputs

TypeNameDescription
Realresult[size(v, 1)]Input vector v normalized to length=1

Function Modelica.​Math.​Vectors.​reverse
Reverse vector elements (e.g., v[1] becomes last element)

Information

Syntax

Vectors.reverse(v);

Description

The function call "Vectors.reverse(v)" returns the vector elements in reverse order.

Example

  reverse({1,2,3,4});  // = {4,3,2,1}

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

Inputs

TypeNameDescription
Realv[:]Real vector

Outputs

TypeNameDescription
Realresult[size(v, 1)]Elements of vector v in reversed order

Function Modelica.​Math.​Vectors.​sort
Sort elements of vector in ascending or descending order

Information

Syntax

           sorted_v = Vectors.sort(v);
(sorted_v, indices) = Vectors.sort(v, ascending=true);

Description

Function sort(..) sorts a Real vector v in ascending order and returns the result in sorted_v. If the optional argument "ascending" is false, the vector is sorted in descending order. In the optional second output argument the indices of the sorted vector with respect to the original vector are given, such that sorted_v = v[indices].

Example

  (v2, i2) := Vectors.sort({-1, 8, 3, 6, 2});
       -> v2 = {-1, 2, 3, 6, 8}
          i2 = {1, 5, 3, 4, 2}

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

Inputs

TypeNameDescription
Realv[:]Real vector to be sorted
Booleanascending= true if ascending order, otherwise descending order

Outputs

TypeNameDescription
Realsorted_v[size(v, 1)]Sorted vector
Integerindices[size(v, 1)]sorted_v = v[indices]

Function Modelica.​Math.​Vectors.​find
Find element in a vector

Information

Syntax

Vectors.find(e, v);
Vectors.find(e, v, eps=0);

Description

The function call "Vectors.find(e, v)" returns the index of the first occurrence of input e in vector v. The test of equality is performed by "abs(e-v[i]) ≤ eps", where "eps" can be provided as third argument of the function. Default is "eps = 0".

Example

  Real v[3] = {1, 2, 3};
  Real e1 = 2;
  Real e2 = 3.01;
  Boolean result;
algorithm
  result := Vectors.find(e1,v);          // = 2
  result := Vectors.find(e2,v);          // = 0
  result := Vectors.find(e2,v,eps=0.1);  // = 3

See also

Vectors.isEqual

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

Inputs

TypeNameDescription
RealeSearch for e
Realv[:]Real vector
RealepsElement e is equal to a element v[i] of vector v if abs(e-v[i]) <= eps

Outputs

TypeNameDescription
Integerresultv[result] = e (first occurrence of e); result=0, if not found

Function Modelica.​Math.​Vectors.​interpolate
Interpolate linearly in a vector

Information

Syntax

// Real    x[:], y[:], xi, yi;
// Integer iLast, iNew;
        yi = Vectors.interpolate(x,y,xi);
(yi, iNew) = Vectors.interpolate(x,y,xi,iLast=1);

Description

The function call "Vectors.interpolate(x,y,xi)" interpolates linearly in vectors (x,y) and returns the value yi that corresponds to xi. Vector x[:] must consist of monotonically increasing values. If xi < x[1] or > x[end], then extrapolation takes places through the first or last two x[:] values, respectively. If the x and y vectors have length 1, then always y[1] is returned. The search for the interval x[iNew] ≤ xi < x[iNew+1] starts at the optional input argument "iLast". The index "iNew" is returned as output argument. The usage of "iLast" and "iNew" is useful to increase the efficiency of the call, if many interpolations take place. If x has two or more identical values then interpolation utilizes the x-value with the largest index.

Example

  Real x1[:] = { 0,  2,  4,  6,  8, 10};
  Real x2[:] = { 1,  2,  3,  3,  4,  5};
  Real y[:]  = {10, 20, 30, 40, 50, 60};
algorithm
  (yi, iNew) := Vectors.interpolate(x1,y,5);  // yi = 35, iNew=3
  (yi, iNew) := Vectors.interpolate(x2,y,4);  // yi = 50, iNew=5
  (yi, iNew) := Vectors.interpolate(x2,y,3);  // yi = 40, iNew=4

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

Inputs

TypeNameDescription
Realx[:]Abscissa table vector (strict monotonically increasing values required)
Realy[size(x, 1)]Ordinate table vector
RealxiDesired abscissa value
IntegeriLastIndex used in last search

Outputs

TypeNameDescription
RealyiOrdinate value corresponding to xi
IntegeriNewxi is in the interval x[iNew] <= xi < x[iNew+1]

Function Modelica.​Math.​Vectors.​relNodePositions
Return vector of relative node positions (0..1)

Information

Syntax

Vectors.relNodePositions(nNodes);

Description

The function call "relNodePositions(nNodes)" returns a vector with the relative positions of the nodes of a discretized pipe with nNodes nodes (including the node at the left and at the right side of the pipe), see next figure:

Example

  Real xsi[7];
algorithm
  xsi = relNodePositions(7);  // xsi = {0, 0.1, 0.3, 0.5, 0.7, 0.9, 1}

See also

MultiBody.Visualizers.PipeWithScalarField

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

Inputs

TypeNameDescription
IntegernNodesNumber of nodes (including node at left and right position)

Outputs

TypeNameDescription
Realxsi[nNodes]Relative node positions