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.
| Links : ± |
CALCULATOR : Polynomial Coefficients From Real Roots Or Zeros [FREE] ±
Calculate polynomial coefficients from the polynomial roots. Enter the roots as a comma separated list, or as separate points on each line. The roots can also be copied and pasted from a spreadsheet. The roots may be entered in any order. There are an infinite number of polynomials with the same roots, so 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. Use the Result Plot option to plot the polymomial value, slope and curvature versus X. Tool Input- R : Polynomial Roots
- An : Nth Coefficient Value
- X : Input X Value
Tool Output- C : Polynomial Curvature At X
- P : Polynomial Value At X
- S : Polynomial Slope At X
|
CALCULATOR : Polynomial Single Real Root Or Zero [FREE] ±
Calculate polynomial real root or zero, maximum or minimum (zero slope), or point of inflection (zero curvature). Enter the polynomial coefficients as a comma separated list, or as separate points on each line. The coefficients can also be copied and pasted from a spreadsheet. The coefficients must be in order A0, A1, A2....An where An is the coefficient for the nth power of x (x^n) etc... 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
- A : Polynomial Coefficients
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)
|
CALCULATOR : Polynomial All Real Roots Or Zeros [FREE] ±
Calculate all polynomial real roots or zeros, maximum and minimum (zero slope), and points of inflection (zero curvature). Enter the polynomial coefficients as a comma separated list, or as separate points on each line. The coefficients can also be copied and pasted from a spreadsheet. The coefficients must be in order A0, A1, A2....An where An is the coefficient for the nth power of x (x^n) etc... The root or zero calculation will cause an error if any of the roots are imaginary, or in some cases with multiple equal roots. Use the single root calculator if there is an error. Imaginary maximum and minimum points, or points of inflection will also cause an error. Use the Result Plot option to plot the polynomial value, slope and curvature. Tool Input- A : Polynomial Coefficients
- X : Input X Value
Tool Output- C : Polynomial Curvature At X
- P : Polynomial Value At X
- S : Polynomial Slope At X
|
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 |
CALCULATOR : Quadratic Polynomial [FREE] ±
Calculate quadratic polynomial value, slope, curvature, real roots or zeros, and maximum or minimum (zero slope). Quadratic second order polynomials have three coefficients. The coefficients must be in order A0, A1, A2 where A2 is the coefficient for the 2nd power of x (x^2) etc... Enter the polynomial coefficients as a comma separated list, or as separate points on each line. The coefficients can also be copied and pasted from a spreadsheet. Quadratics have constant curvature with a single maximum or minimum, and no point of inflection. Use the Result Plot option to plot the polynomial value, slope and curvature. Tool Input- A : Polynomial Coefficients
- X : Input X Value
Tool Output- C : Polynomial Curvature At X
- P : Polynomial Value At X
- S : Polynomial Slope At X
|
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 |
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 |