Vandermonde Matrix Calculator

Calculator

x values

Use commas, spaces, or new lines.

Number of columns

Square matrix when columns equal x count.

Power order

Column ordering affects sign of determinant.

Starting power

Standard Vandermonde starts at 0.

Display precision

Higher precision helps with stability checks.

Determinant method

Determinant shown only for square matrices.

Advanced options

Compute inverse (square only)

Solve coefficients a for V · a = y

Solving is useful for interpolation and fitting.

y values (only if solving)

When enabled, y count must match x count, and the matrix must be square.

Reset

Example data table

Sample input and expected determinant for a standard square Vandermonde.

Example x values	Columns	Order	Start power	Determinant
1, 2, 3, 4	4	Ascending	0	12

Formula used

Matrix entries

For values x₀, …, x_m−1 and columns n, the Vandermonde matrix V is:

V_i,j = x_i^{p + j} (ascending)

V_i,j = x_i^{p + (n − 1 − j)} (descending)

Here p is the starting power.

Determinant (square case)

When m = n and p = 0 with ascending powers:

det(V) = ∏_{0≤i<j≤n−1} (x_j − x_i)

If you reverse the column order (descending), the determinant flips by (−1)^n(n−1)/2. If p > 0, det scales by ∏ x_i^p.

Interpolation link

To fit a polynomial to points (x_i, y_i) with degree n−1, solve:

V · a = y

The coefficient vector a corresponds to the selected column powers and order.

How to use this calculator

Enter your x values using commas, spaces, or new lines.
Choose the number of columns. Match it to x count for square output.
Pick ascending or descending power order, and set starting power.
Optional: enable inverse for conditioning, or enable solving and provide y values.
Press Compute. Download CSV or PDF if needed.

Tip: Vandermonde matrices can become ill-conditioned when x values are close together or when powers get large. Use higher precision when comparing stability.

What it builds

A Vandermonde matrix is a power-basis table built from your x values. Each row corresponds to one sample x_i, and each column is a selected exponent of x_i. In matrix form, polynomial evaluation becomes V·a, so fitting coefficients reduces to solving a linear system. The rank output helps you see whether columns are independent for the chosen inputs and powers.

Power settings that matter

The calculator lets you choose a starting power p and an ascending or descending column order. Starting at p=0 gives the classic form [1, x, x^2, …], while p>0 shifts every exponent upward and scales each row by x_i^p. Descending order reverses columns, which changes coefficient ordering and flips determinant sign for certain sizes. Use the precision control to inspect rounding effects in large powers.

Determinant and uniqueness

For a square matrix with distinct x values, the determinant is nonzero and the interpolation system has a unique solution. In the standard case, det(V)=∏_{i<j}(x_j−x_i). If any two x values repeat, a difference becomes zero and the determinant collapses to zero. When x values are merely close, the product can be tiny, indicating near singularity and unstable coefficients.

Stability and conditioning

Vandermonde matrices can be ill-conditioned when degrees grow, magnitudes explode, or x values cluster. The optional inverse and ∞-norm condition estimate quantify how much input noise can amplify in the solution. As a rule of thumb, a larger condition number means more sensitivity. If conditioning is high, rescale x (for example into [−1,1]), reduce degree, or prefer orthogonal bases like Chebyshev for modeling.

Practical workflows

Use the solve option to obtain polynomial coefficients a from V·a=y, then verify the residual to confirm fit quality against your data. The inverse view is helpful for diagnostics, not just computation. Exporting to CSV supports audits, spreadsheets, and coding pipelines, while the PDF export provides a compact report of inputs, matrix values, determinant, rank, and stability metrics for documentation and sharing. Common applications include polynomial interpolation, curve fitting, signal reconstruction, and error-correcting codes where Vandermonde structure enables fast algorithms in practice.

FAQs

1) Why is the determinant not shown?

The determinant is defined only for square matrices. Make the number of columns equal to the count of x values, then recompute to see the determinant and related diagnostics.

2) What does the rank value mean?

Rank measures how many independent columns the matrix has. Full rank suggests the columns span the space for your chosen powers, while reduced rank signals repeated x values, insufficient variation, or numerical degeneracy.

3) When should I choose descending powers?

Choose descending order when you want coefficients listed from highest degree down to the starting power. The matrix values are the same set of columns, just reversed, which can change determinant sign and coefficient indexing.

4) Why can the inverse fail to compute?

If the matrix is singular or extremely ill-conditioned, pivoting may detect a near-zero pivot and stop. Try distinct x values, reduce columns, rescale x, or increase display precision to inspect instability.

5) How does the solve option relate to interpolation?

With solve enabled, enter y values matching each x. For a square matrix, the solver finds coefficients a so V·a=y, yielding a polynomial in the selected power basis that passes through your points.

6) How should I set columns and precision?

Columns control the polynomial degree and matrix width. For classic interpolation use columns equal to x count. Increase precision when x values are close or powers are large, because rounding can hide small differences that affect rank and conditioning.