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Polynomials Modules

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CALCULATOR MODULE : Maths Polynomial   ±

Calculate polynomial coefficients, roots or zeros, maximum and minimum, points of inflection, and interpolate polynomial value, slope and curvature.

Polynomials can be calculated for linear (first order), quadratic (second order), cubic (third order), quartic (fourth order), quintic (fifth order), sextic (sixth order), septic (seventh order), octic (eighth order) or nth degree. For polynomials with all real roots, all roots can sometimes be solved simultaneously using the Durand Kerner method. In other cases solve for individual roots. The maximum or minimum points (slope equals zero) and the inflection points (curvature equals zero) can also be calculated. Use a plot page to plot the polynomial and identify the approximate root values if any.

Lagrange's method is used to interpolate between data points. This method is useful for interpolating between data points, but can give poor results when extrapolating outside the data range. Evenly spaced data points can result in cyclic behaviour.

Polynomial coefficients can be calculated from the real roots, and the nth coefficient. There are an infinite number of polynomials with the same roots. The nth coefficient is required in order to calculate unique coefficients. This method only applies if all of the roots are real. Polynomial coefficients can also be calculated from XZ data points.

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CALCULATOR MODULE : Maths Polynomial Coefficient   ±

Calculate polynomial coefficients from real roots or zeros, or from XZ data points.

To calculate the coeficients from real roots, the value of An (the nth order coefficient) must be included in order to calculate a unique set of polynomial coefficients. The coefficients are listed in the order A0, A1, A2....An where An is the coefficient for the nth power of x (x^n) etc... The calculation is only valid for polynomials with no imaginary roots, ie the number of real roots equals the polynomial order.

The polynomial value, slope and curvature are calculated at X. The maximum and minimum points (zero slope), and the points of inflection (zero curvature), can not be calculated unless all points are real numbers.

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CALCULATOR MODULE : Maths Polynomial Root Maximum And Minimum And Point Of Inflection   ±

Calculate maths polynomial roots, maximum and minimum and points of inflection.

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. Use the single root calculators for polynomials with imaginary roots. The single root calculators can only calculate real roots, maximum and minimum or points of inflection. The value Xu is used as the initial guess for calculating the roots, maximum and minimum or points of inflection. Use the plot options to select suitable guess values.

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CALCULATOR MODULE : 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.

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CALCULATOR MODULE : Maths Lagrange Polynomial Interpolation   ±

Calculate a Lagrange polynomial to fit the data points. The polynomial passes through all the data points.

Higher order polynomials may be cyclic (lower order polynomials are preferred). The slope and curvature are not continuous. Extrapolated values should be used carefully.

For vector data calculators enter the data as X,Z pairs separated by a comma or tab, with each pair on a new line. Or copy and paste the data points from a spreadsheet. For array data calculators, enter the X and Z values as separate comma or tab separated lists. Use the data plot option on the plot bar to display the data points and the Lagrange polynomial.

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