Library of functions operating on vectors
This library provides functions operating on vectors:
Extends from Modelica.Icons.Package (Icon for standard packages).
Name | Description |
---|---|
toString | Convert a real vector in to a string representation |
isEqual | Determine if two Real vectors are numerically identical |
norm | Return the p-norm of a vector |
length | Return length of a vector (better as norm(), if further symbolic processing is performed) |
normalize | Return normalized vector such that length = 1 and prevent zero-division for zero vector |
normalizeWithAssert | Return normalized vector such that length = 1 (trigger an assert for zero vector) |
reverse | Reverse vector elements (e.g., v[1] becomes last element) |
sort | Sort elements of vector in ascending or descending order |
find | Find element in a vector |
interpolate | Interpolate linearly in a vector |
relNodePositions | Return vector of relative node positions (0..1) |
Utilities | Utility functions that should not be directly utilized by the user |
Convert a real vector in to a string representation
Vectors.toString(v); Vectors.toString(v,name="",significantDigits=6);
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.
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"
Extends from Modelica.Icons.Function (Icon for functions).
Name | Description |
---|---|
v[:] | Real vector |
name | Independent variable name used for printing |
significantDigits | Number of significant digits that are shown |
Name | Description |
---|---|
s |
Determine if two Real vectors are numerically identical
Vectors.isEqual(v1, v2); Vectors.isEqual(v1, v2, eps=0);
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".
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
Vectors.find, Matrices.isEqual, Strings.isEqual
Extends from Modelica.Icons.Function (Icon for functions).
Name | Description |
---|---|
v1[:] | First vector |
v2[:] | Second vector (may have different length as v1) |
eps | Two elements e1 and e2 of the two vectors are identical if abs(e1-e2) <= eps |
Name | Description |
---|---|
result | = true, if vectors have the same length and the same elements |
Return the p-norm of a vector
Vectors.norm(v); Vectors.norm(v,p=2); // 1 ≤ p ≤ ∞
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:
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)
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
Extends from Modelica.Icons.Function (Icon for functions).
Name | Description |
---|---|
v[:] | Real vector |
p | Type of p-norm (often used: 1, 2, or Modelica.Constants.inf) |
Name | Description |
---|---|
result | p-norm of vector v |
Return length of a vector (better as norm(), if further symbolic processing is performed)
Vectors.length(v);
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(..).
v = {2, -4, -2, -1}; length(v); // = 5
Extends from Modelica.Icons.Function (Icon for functions).
Name | Description |
---|---|
v[:] | Real vector |
Name | Description |
---|---|
result | Length of vector v |
Return normalized vector such that length = 1 and prevent zero-division for zero vector
Vectors.normalize(v); Vectors.normalize(v,eps=100*Modelica.Constants.eps);
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.
normalize({1,2,3}); // = {0.267, 0.534, 0.802} normalize({0,0,0}); // = {0,0,0}
Vectors.length, Vectors.normalizeWithAssert
Extends from Modelica.Icons.Function (Icon for functions).
Name | Description |
---|---|
v[:] | Real vector |
eps | if |v| < eps then result = v/eps |
Name | Description |
---|---|
result[size(v, 1)] | Input vector v normalized to length=1 |
Return normalized vector such that length = 1 (trigger an assert for zero vector)
Vectors.normalizeWithAssert(v);
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.
normalizeWithAssert({1,2,3}); // = {0.267, 0.534, 0.802} normalizeWithAssert({0,0,0}); // error (an assert is triggered)
Vectors.length, Vectors.normalize
Extends from Modelica.Icons.Function (Icon for functions).
Name | Description |
---|---|
v[:] | Real vector |
Name | Description |
---|---|
result[size(v, 1)] | Input vector v normalized to length=1 |
Reverse vector elements (e.g., v[1] becomes last element)
Vectors.reverse(v);
The function call "Vectors.reverse(v)
" returns the
vector elements in reverse order.
reverse({1,2,3,4}); // = {4,3,2,1}
Extends from Modelica.Icons.Function (Icon for functions).
Name | Description |
---|---|
v[:] | Real vector |
Name | Description |
---|---|
result[size(v, 1)] | Elements of vector v in reversed order |
Sort elements of vector in ascending or descending order
sorted_v = Vectors.sort(v); (sorted_v, indices) = Vectors.sort(v, ascending=true);
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].
(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).
Name | Description |
---|---|
v[:] | Real vector to be sorted |
ascending | = true if ascending order, otherwise descending order |
Name | Description |
---|---|
sorted_v[size(v, 1)] | Sorted vector |
indices[size(v, 1)] | sorted_v = v[indices] |
Find element in a vector
Vectors.find(e, v); Vectors.find(e, v, eps=0);
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".
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
Extends from Modelica.Icons.Function (Icon for functions).
Name | Description |
---|---|
e | Search for e |
v[:] | Real vector |
eps | Element e is equal to a element v[i] of vector v if abs(e-v[i]) <= eps |
Name | Description |
---|---|
result | v[result] = e (first occurrence of e); result=0, if not found |
Interpolate linearly in a vector
// Real x[:], y[:], xi, yi; // Integer iLast, iNew; yi = Vectors.interpolate(x,y,xi); (yi, iNew) = Vectors.interpolate(x,y,xi,iLast=1);
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.
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).
Name | Description |
---|---|
x[:] | Abscissa table vector (strict monotonically increasing values required) |
y[size(x, 1)] | Ordinate table vector |
xi | Desired abscissa value |
iLast | Index used in last search |
Name | Description |
---|---|
yi | Ordinate value corresponding to xi |
iNew | xi is in the interval x[iNew] <= xi < x[iNew+1] |
Return vector of relative node positions (0..1)
Vectors.relNodePositions(nNodes);
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:
Real xsi[7]; algorithm xsi = relNodePositions(7); // xsi = {0, 0.1, 0.3, 0.5, 0.7, 0.9, 1}
MultiBody.Visualizers.PipeWithScalarField
Extends from Modelica.Icons.Function (Icon for functions).
Name | Description |
---|---|
nNodes | Number of nodes (including node at left and right position) |
Name | Description |
---|---|
xsi[nNodes] | Relative node positions |