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Maths Spline Modules

Keyword(s) Maths Spline. For a new search enter search key word(s) then click GO GO

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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 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 Linear Regression   ±

Calculate the best fit line for scatter data points using the least squares linear regression method. The curve does not have to pass through each data point.

For straight line or linear curves (Z = A x + B) the regression is performed directly on the X and Z data values. For power curves (Z = A x^B) the regression is performed on the ln(X) and ln(Z) values. For logarithmic curves (Z = ln(X)) the regression is performed on the ln(X) and Z values. For exponential curves (Z = A e^B) the regression is performed on the X and ln(Z) values. For the user defined transform (Z = A f(X) + B) the regression is performed on f(X) and Z where f(X) is the user defined transform.

The X and Z offsets can be used to change the origin for log values (ln(X - Xo) and ln(Z - Zo)) and user defined transform (f(X - Xo)). The offsets are not used for the X and Z values.

The Z unit value is applied for the log of negative Z values. The Z unit value is not applied for X and Z values, or for user defined transforms (user defined transforms should account for the sign of the data points).

The regression data and regression parameters are displayed in the output view at the bottom of the page. The correlation coefficient r is a measure of how well the curve fits the data points (close to one is better). Extrapolated values should be used carefully.

Enter vector 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. Enter array data X and Z values as separate comma or tab separated lists. Store file data to a text file as comma or tab separated pairs (X,Z), with each pair on a new line (or copy and past cells from a spreadsheet). Refer to the example text file in resources.

Use the data plot option on the plot bar to display the data points and the best fit line.

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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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CALCULATOR MODULE : Maths Cubic Spline   ±

Calculate a cubic spline to fit the data points.

The cubic spline is calculated so that the slope and curvature match at each data point. The end points are assumed to have zero curvature. 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 b(u, m)ar to display the data points and the cubic spline.

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CALCULATOR MODULE : Maths Curve Data Check   ±

Calculate smoothed data values from raw input data.

The calculator is intended for checking and smoothing digitised plot data. Use the Result Plot option to display the data value, slope and curvature. The value, slope and curvature curves should be smooth. Sudden changes in slope and or curvature indicate possible faulty values.

Options include data order (ascending or descending data), plot axies (Z versus X or X versus Z), smoothing type, and whether to smooth the maximum and minimum values. Smoothing uses a simple weighted mean value.

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CALCULATOR MODULE : Weibull Gumbel And Frechet Extreme Event Probability   ±

Calculate extreme event amplitude and return period from return period data using the Weibull, Gumbel and Frechet probability distributions.

A best fit line is calculated for the data points using the least squares linear regression method. The regression is calculated for X versus Z instead of Z versus X (the X and Z values are swapped). The three parameter distribution amplitude offset is a minimum amplitude. The regression data points and regression parameters are displayed in the output view at the bottom of the page. Use the Data Plot option on the plot bar to display the data points and the best fit line.

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