## 70 Package interpol ¶

### 70.1 Introduction to interpol ¶

Package `interpol` defines the Lagrangian, the linear and the cubic splines methods for polynomial interpolation.

For comments, bugs or suggestions, please contact me at ’mario AT edu DOT xunta DOT es’.

Categories: Numerical methods · Share packages · Package interpol ·

### 70.2 Functions and Variables for interpol ¶

Function: lagrange
lagrange (points)
lagrange (points, option) ¶

  Computes the polynomial interpolation by the Lagrangian method. Argument points must be either:

  - a two column matrix, `p:matrix([2,4],[5,6],[9,3])`,
  - a list of pairs, `p: [[2,4],[5,6],[9,3]]`,
  - a list of numbers, `p: [4,6,3]`, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.

  In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

  With the option argument it is possible to select the name for the independent variable, which is `'x` by default; to define another one, write something like `varname='z`.

  Note that when working with high degree polynomials, floating point evaluations are unstable.

  See also `linearinterpol`, `cspline`, and `ratinterpol`.

  Examples:

      (%i1) load("interpol")$
      (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
      (%i3) lagrange(p);
             (x - 7) (x - 6) (x - 3) (x - 1)
      (%o3)  -------------------------------
                           35
         (x - 8) (x - 6) (x - 3) (x - 1)
       - -------------------------------
                       12
         7 (x - 8) (x - 7) (x - 3) (x - 1)
       + ---------------------------------
                        30
         (x - 8) (x - 7) (x - 6) (x - 1)
       - -------------------------------
                       60
         (x - 8) (x - 7) (x - 6) (x - 3)
       + -------------------------------
                       84
      (%i4) f(x):=''%;
                     (x - 7) (x - 6) (x - 3) (x - 1)
      (%o4)  f(x) := -------------------------------
                                   35
         (x - 8) (x - 6) (x - 3) (x - 1)
       - -------------------------------
                       12
         7 (x - 8) (x - 7) (x - 3) (x - 1)
       + ---------------------------------
                        30
         (x - 8) (x - 7) (x - 6) (x - 1)
       - -------------------------------
                       60
         (x - 8) (x - 7) (x - 6) (x - 3)
       + -------------------------------
                       84
      (%i5) /* Evaluate the polynomial at some points */
            expand(map(f,[2.3,5/7,%pi]));
                                        4          3           2
                          919062  73 %pi    701 %pi    8957 %pi
      (%o5)  [- 1.567535, ------, ------- - -------- + ---------
                          84035     420       210         420
                                                   5288 %pi   186
                                                 - -------- + ---]
                                                     105       5
      (%i6) %,numer;
      (%o6) [- 1.567535, 10.9366573451538, 2.89319655125692]
      (%i7) load("draw")$  /* load draw package */
      (%i8) /* Plot the polynomial together with points */
            draw2d(
              color      = red,
              key        = "Lagrange polynomial",
              explicit(f(x),x,0,10),
              point_size = 3,
              color      = blue,
              key        = "Sample points",
              points(p))$
      (%i9) /* Change variable name */
            lagrange(p, varname=w);
             (w - 7) (w - 6) (w - 3) (w - 1)
      (%o9)  -------------------------------
                           35
         (w - 8) (w - 6) (w - 3) (w - 1)
       - -------------------------------
                       12
         7 (w - 8) (w - 7) (w - 3) (w - 1)
       + ---------------------------------
                        30
         (w - 8) (w - 7) (w - 6) (w - 1)
       - -------------------------------
                       60
         (w - 8) (w - 7) (w - 6) (w - 3)
       + -------------------------------
                       84

  Categories: Package interpol ·

Function: charfun2 (x, a, b) ¶

  The characteristic or indicator function on the half-open interval \([\mathit{a}, \mathit{b})\), that is, including a and excluding b.

  When \(\mathit{x} >= \mathit{a}\) and \(\mathit{x} < \mathit{b}\) evaluates to `true` or `false`, `charfun2` returns 1 or 0, respectively.

  Otherwise, `charfun2` returns a partially-evaluated result in terms of `charfun`.

  Package `interpol` loads this function.

  See also `charfun`.

  Examples:

  When \(\mathit{x} >= \mathit{a}\) and \(\mathit{x} < \mathit{b}\) evaluates to `true` or `false`, `charfun2` returns 1 or 0, respectively.

      (%i1) load ("interpol") $
      (%i2) charfun2 (5, 0, 100);
      (%o2)                           1
      (%i3) charfun2 (-5, 0, 100);
      (%o3)                           0

  Otherwise, `charfun2` returns a partially-evaluated result in terms of `charfun`.

      (%i1) load ("interpol") $
      (%i2) charfun2 (t, 0, 100);
      (%o2)            charfun((0 <= t) and (t < 100))
      (%i3) charfun2 (5, u, v);
      (%o3)             charfun((u <= 5) and (5 < v))
      (%i4) assume (v > u, u > 5);
      (%o4)                    [v > u, u > 5]
      (%i5) charfun2 (5, u, v);
      (%o5)                           0

  Categories: Package interpol ·

Function: linearinterpol
linearinterpol (points)
linearinterpol (points, option) ¶

  Computes the polynomial interpolation by the linear method. Argument points must be either:

  - a two column matrix, `p:matrix([2,4],[5,6],[9,3])`,
  - a list of pairs, `p: [[2,4],[5,6],[9,3]]`,
  - a list of numbers, `p: [4,6,3]`, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.

  In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

  With the option argument it is possible to select the name for the independent variable, which is `'x` by default; to define another one, write something like `varname='z`.

  See also `lagrange`, `cspline`, and `ratinterpol`.

  Examples:

      (%i1) load("interpol")$
      (%i2) p: matrix([7,2],[8,3],[1,5],[3,2],[6,7])$
      (%i3) linearinterpol(p);
              13   3 x
      (%o3)  (-- - ---) charfun2(x, minf, 3)
              2     2
       + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
          5 x
       + (--- - 3) charfun2(x, 3, 6)
           3

      (%i4) f(x):=''%;
                      13   3 x
      (%o4)  f(x) := (-- - ---) charfun2(x, minf, 3)
                      2     2
       + (x - 5) charfun2(x, 7, inf) + (37 - 5 x) charfun2(x, 6, 7)
          5 x
       + (--- - 3) charfun2(x, 3, 6)
           3
      (%i5)  /* Evaluate the polynomial at some points */
             map(f,[7.3,25/7,%pi]);
                                  62  5 %pi
      (%o5)                 [2.3, --, ----- - 3]
                                  21    3
      (%i6) %,numer;
      (%o6)  [2.3, 2.952380952380953, 2.235987755982989]
      (%i7) load("draw")$  /* load draw package */
      (%i8)  /* Plot the polynomial together with points */
             draw2d(
               color      = red,
               key        = "Linear interpolator",
               explicit(f(x),x,-5,20),
               point_size = 3,
               color      = blue,
               key        = "Sample points",
               points(args(p)))$
      (%i9)  /* Change variable name */
             linearinterpol(p, varname='s);
             13   3 s
      (%o9) (-- - ---) charfun2(s, minf, 3)
             2     2
       + (s - 5) charfun2(s, 7, inf) + (37 - 5 s) charfun2(s, 6, 7)
          5 s
       + (--- - 3) charfun2(s, 3, 6)
           3

  Categories: Package interpol ·

Function: cspline
cspline (points)
cspline (points, option1, option2, ...) ¶

  Computes the polynomial interpolation by the cubic splines method. Argument points must be either:

  - a two column matrix, `p:matrix([2,4],[5,6],[9,3])`,
  - a list of pairs, `p: [[2,4],[5,6],[9,3]]`,
  - a list of numbers, `p: [4,6,3]`, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.

  In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

  There are three options to fit specific needs:

  - `'d1`, default `'unknown`, is the first derivative at \(x_1\); if it is `'unknown`, the second derivative at \(x_1\) is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
  - `'dn`, default `'unknown`, is the first derivative at \(x_n\); if it is `'unknown`, the second derivative at \(x_n\) is made equal to 0 (natural cubic spline); if it is equal to a number, the second derivative is calculated based on this number.
  - `'varname`, default `'x`, is the name of the independent variable.

  See also `lagrange`, `linearinterpol`, and `ratinterpol`.

  Examples:

      (%i1) load("interpol")$
      (%i2) p:[[7,2],[8,2],[1,5],[3,2],[6,7]]$
      (%i3) /* Unknown first derivatives at the extremes
               is equivalent to natural cubic splines */
            cspline(p);
                    3         2
              1159 x    1159 x    6091 x   8283
      (%o3)  (------- - ------- - ------ + ----) charfun2(x, minf, 3)
               3288      1096      3288    1096
                  3         2
            2587 x    5174 x    494117 x   108928
       + (- ------- + ------- - -------- + ------) charfun2(x, 7, inf)
             1644       137       1644      137
                3          2
          4715 x    15209 x    579277 x   199575
       + (------- - -------- + -------- - ------) charfun2(x, 6, 7)
           1644       274        1644      274
                  3         2
            3287 x    2223 x    48275 x   9609
       + (- ------- + ------- - ------- + ----) charfun2(x, 3, 6)
             4932       274      1644     274

      (%i4) f(x):=''%$
      (%i5) /* Some evaluations */
            map(f,[2.3,5/7,%pi]), numer;
      (%o5) [1.991460766423356, 5.823200187269903, 2.227405312429507]
      (%i6) load("draw")$  /* load draw package */
      (%i7) /* Plotting interpolating function */
            draw2d(
              color      = red,
              key        = "Cubic splines",
              explicit(f(x),x,0,10),
              point_size = 3,
              color      = blue,
              key        = "Sample points",
              points(p))$
      (%i8) /* New call, but giving values at the derivatives */
            cspline(p,d1=0,dn=0);
                    3          2
              1949 x    11437 x    17027 x   1247
      (%o8)  (------- - -------- + ------- + ----) charfun2(x, minf, 3)
               2256       2256      2256     752
                  3          2
            1547 x    35581 x    68068 x   173546
       + (- ------- + -------- - ------- + ------) charfun2(x, 7, inf)
              564       564        141      141
               3          2
          607 x    35147 x    55706 x   38420
       + (------ - -------- + ------- - -----) charfun2(x, 6, 7)
           188       564        141      47
                  3         2
            3895 x    1807 x    5146 x   2148
       + (- ------- + ------- - ------ + ----) charfun2(x, 3, 6)
             5076       188      141      47
      (%i8) /* Defining new interpolating function */
            g(x):=''%$
      (%i9) /* Plotting both functions together */
            draw2d(
              color      = black,
              key        = "Cubic splines (default)",
              explicit(f(x),x,0,10),
              color      = red,
              key        = "Cubic splines (d1=0,dn=0)",
              explicit(g(x),x,0,10),
              point_size = 3,
              color      = blue,
              key        = "Sample points",
              points(p))$

  Categories: Package interpol ·

Function: ratinterpol
ratinterpol (points, numdeg)
ratinterpol (points, numdeg, option1) ¶

  Generates a rational interpolator for data given by points and the degree of the numerator being equal to numdeg; the degree of the denominator is calculated automatically. Argument points must be either:

  - a two column matrix, `p:matrix([2,4],[5,6],[9,3])`,
  - a list of pairs, `p: [[2,4],[5,6],[9,3]]`,
  - a list of numbers, `p: [4,6,3]`, in which case the abscissas will be assigned automatically to 1, 2, 3, etc.

  In the first two cases the pairs are ordered with respect to the first coordinate before making computations.

  There is one option to fit specific needs:

  - `'varname`, default `'x`, is the name of the independent variable.

  See also `lagrange`, `linearinterpol`, `cspline`, `minpack_lsquares`, and Package lbfgs

  Examples:

      (%i1) load("interpol")$
      (%i2) load("draw")$
      (%i3) p:[[7.2,2.5],[8.5,2.1],[1.6,5.1],[3.4,2.4],[6.7,7.9]]$
      (%i4) for k:0 thru length(p)-1 do
              draw2d(
                explicit(ratinterpol(p,k),x,0,9),
                point_size = 3,
                points(p),
                title = concat("Degree of numerator = ",k),
                yrange=[0,10])$

  Categories: Package interpol ·

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