Last Updated on January 13, 2026 by Maged kamel
Quadratic interpolation.
Quadratic interpolation is using a second-order polynomial to make interpolation for a function.
Unlike linear interpolation, which we discussed in the previous post, quadratic interpolation requires three points.
The first point of the three points has a coordinate of(x0,y0 ), the second point has a coordinate of(x1,y1 ), and the last point has a coordinate of(x2,y2 )
What is the Vandermonde matrix?
After substituting the polynomial matrix with the values of the different x and y of the given three points, we will have a set of quadratic equations with three unknowns, a0, a1, and a2. These equations can be written in a matrix form. The Form of V*X=Y is to be used, where V is the Vandermonde matrix, Xs is the column vector for the coefficients of a0,a1, and a2.y is the column vector of y values for the three points.
How do we derive the expression for the Vandermonde elements?
The Vandermonde matrix, in the case of quadratic polynomials, is a(3 x 3) matrix, it can be written in the form of Vi,j =xi-1^j-1, where i is the row number and j is the column number, for instance,, V23=x2-1^j(3-1)=x1^2, V23 is the element in the second row and the third column, will be equal to the second power of x1.
I have written the Vandermonde matrix in that form for quadratic interpolation.
How do we find the determinant value of the Vandermonde matrix?
To get the matrix’s determinant, we set x0=0 by multiplying the second column by x0, subtracting the result from the third column, and placing the result in the third column.
This is a continuation of the calculations.
To get the determinant of the matrix, we will let x0=0 by multiplying the second column by x0, then subtracting the result from the third column and placing the result in the third column.
The process will be repeated by the first column by x0, then the new result of that multiplication will be subtracted from the second column and placed in the second column.
The first row will have two zeros in the second and third columns, while the second row will have a value in the first column.
Finally, the first column will have a value in the third row.
The final value of the determinant of the Vandermonde matrix can be found as (x2-x0)*(x1-x0)*(x2-x1).
The process to get the inverse of the Vandermonde matrix.
We want to find the inverse matrix of the Vandermonde matrix, which is done using cofactors and the adjugate.
We will start estimating the minors of matrix V. The next images will show the estimation process for the minors of the first, second, and third rows.
These are the values of the minors for the second row of the matrix. The values of the minors are for the third row of the matrix.
The step-by-step procedure is shown in the next slide image.
We can form the co-actor matrix and factor (x2-x1) for the first column & (x2-x0) for the second column, and (x2-x1) for the last column.
To get the inverse of the V—matrix, we will divide the cofactor matrix by the determinant value, which we estimated earlier.
That division cancels the common factor, and the final result of the inverse matrix can be found.
The final value of the coefficient matrix.
The last step is to multiply the inverse matrix V-1 by the X-X-matrix to find the value of the factor column vector.
In the end, we obtained these values, as shown in the last slide image.
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The next post: Solved problems for quadratic interpolation.