This information is part of the Modelica Standard Library maintained by the Modelica Association.

Univariate constant, linear or cubic Hermite spline interpolation in one dimension of a table. Via parameter columns it can be defined how many columns of the table are interpolated. If, e.g., columns={2,4}, it is assumed that 2 output signals are present and that the first output interpolates via column 2 and the second output interpolates via column 4 of the table matrix.

The grid points and function values are stored in a matrix "table[i,j]", where the first column "table[:,1]" contains the grid points and the other columns contain the data to be interpolated. Example:

table = [0,  0;
         1,  1;
         2,  4;
         4, 16]
If, e.g., the input u = 1.0, the output y =  1.0,
    e.g., the input u = 1.5, the output y =  2.5,
    e.g., the input u = 2.0, the output y =  4.0,
    e.g., the input u =-1.0, the output y = -1.0 (i.e., extrapolation).

• The interpolation interval is found by a binary search where the interval used in the last call is used as start interval.
• Via parameter smoothness it is defined how the data is interpolated:

  smoothness = 1: Linear interpolation
             = 2: Akima interpolation: Smooth interpolation by cubic Hermite
                  splines such that der(y) is continuous, also if extrapolated.
             = 3: Constant segments
             = 4: Fritsch-Butland interpolation: Smooth interpolation by cubic
                  Hermite splines such that y preserves the monotonicity and
                  der(y) is continuous, also if extrapolated.
             = 5: Steffen interpolation: Smooth interpolation by cubic Hermite
                  splines such that y preserves the monotonicity and der(y)
                  is continuous, also if extrapolated.
             = 6: Modified Akima interpolation: Smooth interpolation by cubic
                  Hermite splines such that der(y) is continuous, also if
                  extrapolated. Additionally, overshoots and edge cases of the
                  original Akima interpolation method are avoided.
  

• First and second derivatives are provided, with exception of the following two smoothness options.
  1. No derivatives are provided for interpolation by constant segments.
  2. No second derivative is provided for linear interpolation.
• Values outside of the table range, are computed by extrapolation according to the setting of parameter extrapolation:

  extrapolation = 1: Hold the first or last value of the table,
                     if outside of the table scope.
                = 2: Extrapolate by using the derivative at the first/last table
                     points if outside of the table scope.
                     (If smoothness is LinearSegments or ConstantSegments
                     this means to extrapolate linearly through the first/last
                     two table points.).
                = 3: Periodically repeat the table data (periodical function).
                = 4: No extrapolation, i.e. extrapolation triggers an error
  

• If the table has only one row, the table value is returned, independent of the value of the input signal.
• The grid values (first column) have to be strictly increasing.

The table matrix can be defined in the following ways:

1. Explicitly supplied as parameter matrix "table", and the other parameters have the following values:

   tableName is "NoName" or has only blanks,
   fileName  is "NoName" or has only blanks.
   

2. Read from a file "fileName" where the matrix is stored as "tableName". Both text and MATLAB MAT-file format is possible. (The text format is described below). The MAT-file format comes in four different versions: v4, v6, v7 and v7.3. The library supports at least v4, v6 and v7 whereas v7.3 is optional. It is most convenient to generate the MAT-file from FreeMat or MATLAB® by command

   save tables.mat tab1 tab2 tab3
   

   or Scilab by command

   savematfile tables.mat tab1 tab2 tab3
   

   when the three tables tab1, tab2, tab3 should be used from the model.
   Note, a fileName can be defined as URI by using the helper function loadResource.
3. Statically stored in function "usertab" in file "usertab.c". The matrix is identified by "tableName". Parameter fileName = "NoName" or has only blanks. Row-wise storage is always to be preferred as otherwise the table is reallocated and transposed. See the Tables package documentation for more details.

When the constant "NO_FILE_SYSTEM" is defined, all file I/O related parts of the source code are removed by the C-preprocessor, such that no access to files takes place.

If tables are read from a text file, the file needs to have the following structure ("-----" is not part of the file content):

-----------------------------------------------------
#1
double tab1(5,2)   # comment line
  0   0
  1   1
  2   4
  3   9
  4  16
double tab2(5,2)   # another comment line
  0   0
  2   2
  4   8
  6  18
  8  32
-----------------------------------------------------

Note, that the first two characters in the file need to be "#1" (a line comment defining the version number of the file format). Afterwards, the corresponding matrix has to be declared with type (= "double" or "float"), name and actual dimensions. Finally, in successive rows of the file, the elements of the matrix have to be given. The elements have to be provided as a sequence of numbers in row-wise order (therefore a matrix row can span several lines in the file and need not start at the beginning of a line). Numbers have to be given according to C syntax (such as 2.3, -2, +2.e4). Number separators are spaces, tab (\t), comma (,), or semicolon (;). Several matrices may be defined one after another. Line comments start with the hash symbol (#) and can appear everywhere. Text files should either be ASCII or UTF-8 encoded, where UTF-8 encoded strings are only allowed in line comments and an optional UTF-8 BOM at the start of the text file is ignored. Other characters, like trailing non comments, are not allowed in the file.

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