Maths Polynomial Interpolation

Calculate polynomial value, slope and curvature by interpolating between data points.

Lagrange polynomial interpolation is calculated by breaking up the data into sections and using a linear, quadratic, cubic, quartic, or quintic polynomial for each section. The cubic interpolation is recommended for most situations. The nth degree polynomial option uses all of the data points in a single section. The Lagrange method calculates the polynommial value only (not the slope or curvature), and does not calculate the polynomial coefficients, roots, maximum and minimum, or points of inflection.

Polynomial coefficients can be calculated from XZ data points using matrix factorisation using all of the data points. The polynomial value, slope and curvature are then calculated from the coefficients.

Cubic splines are used to fit a cubic polynomial between each set of XZ data points. The cubic spline method calculates the polynommial value only (not the slope or curvature), and does not calculate the polynomial coefficients, roots, maximum and minimum, or points of inflection.

Note : Using high order polynomials may cause harmonics on the interpolation line, particularly if the data points are evenly spaced. Extrapolated results calculated from outside the range of data points should be used very carefully, particularly for high order polynomials. Use the plot options to check the behaviour of the polynomial outside the data range.

Multiple roots, maximum and minimum or points of inflection are calculated using the Durand Kerner method. This method is only valid if all roots, maximum and minimum or points of inflection are real.

Change Module :

Maths Polynomial Calculation Module
Maths Polynomial Coefficient Calculation Module
Maths Polynomial Root Maximum And Minimum And Point Of Inflection Calculation Module
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CALCULATOR : Polynomial Coefficients And Interpolation From Data Points [FREE] ±

Calculate polynomial coefficients from all polynomial data points using matrix factorisation.

Enter data as comma separated pairs (X,Z), with each pair on a new line, or copy and past cells from a spreadsheet. The data should be in ascending order.

The polynomial value, slope and curvature are calculated at either the user defined X, the root or zero, the maximum or minimum, or the point of inflection. The maximum and minimum can only be calculated for polynomials of second order or greater. The point of inflection can only be calculated for polynomials of third order or greater. The root, maximum or minimum, or point of inflection must be real.

The user defined Xu value is used as the initial guess value for calculating the root and other points. Use the Result Plot option to plot the polymomial value, slope and curvature versus X, and to determine suitable guess values for Xu.

Tool Input

ptype : Polynomial Calculation Type
- Xu : User Defined X Value or Guess Value
Xdata : X Data Values
Zdata : Z Data Values

Tool Output

C : Polynomial Curvature At X
P : Polynomial Value At X
S : Polynomial Slope At X
X : X Value (Input, Root, Maximum or Minimum, Or Inflection Point)
cvg : Convergence (== 1)
|M| : Matrix Determinant (!= 0)

CALCULATOR : Lagrange Polynomial Interpolation From Data Points [FREE] ±

Interpolate between data points using a Lagrange polynomial interpolation.

Enter data as comma separated pairs (X,Z), with each pair on a new line, or copy and past cells from a spreadsheet. The interpolation can be performed either by breaking up the data into smaller sections and using a linear, quadratic, cubic, quartic, or quintic polynomial for each section, or by using an nth degree polynomial which uses all of the data points. The cubic interpolation is recommended for most situations. The calculator will automatically change the polynommial type if there are not enough data points. The Lagrange method does not calculate the polynomial coefficients.

Note : Using high order polynomials may cause harmonics on the interpolation line, particularly if the data points are evenly spaced. Extrapolated results calculated from outside the range of data points should be used very carefully, particularly for high order polynomials. Use the Data Plot option to plot the data points, and to check the interpolation line.

Tool Input

ntype : Number Of Polynomial Terms
OOR : Out Of Range Extrapolation Type
Xdata : X Data Values
Zdata : Z Data Values
X : X Value

Tool Output

Z : Z Value

CALCULATOR : Cubic Spline Interpolation From Data Points [FREE] ±

Calculate cubic spline interpolation with natural end conditions (zero bending moment at the end points) from data points.

Enter data as comma separated pairs (X,Z), with each pair on a new line (or copy and past cells from a spreadsheet). A minimum of four data points are required for a spline (a linear curve is used if there are less than four data points). Use the Data Plot option to plot the data points and the curve.

Tool Input

OOR : Out Of Range Type
Xdata : X Data
Zdata : Z Data
X : X Input Value

Tool Output

Z : Z Output Value

CALCULATOR : Two Axis Linear Cubic Spline And Polynomial Interpolation From Data Points [FREE] ±

Calculate two axis curve from data points using either linear, cubic spline, or polynomial interpolation.

Enter the data (Z data) as comma separted rows, with each row on a new line (or copy and past cells from a spreadsheet). Enter the I row values (Idata) and J column values (Jdata) as comma separated lists. The value Zij is the Z value from the ith row and the jth column. Use the Data Plot option to plot the data points and the caculated curve.

Tool Input

ntype : Number Of Polynomial Terms
oor : Out Of Range Type
optype : Surface Type
Zdata : Surface Data
X : X Input Value
Y : Y Input Value

Tool Output

Z : Z Output Value