Vandermonde Determinant 3 Calculator

Compute oriented determinant of rows [1,x,x²] from three nodes without requiring caller prebuilt matrix; repeated nodes yield exact singular result.

Description

A Vandermonde matrix encodes nodes as powers in successive columns.

A Vandermonde matrix encodes nodes as powers in successive columns. Node order controls determinant sign, while repeated nodes make the matrix singular.^[1]

Compute oriented determinant of rows [1,x,x²] from three nodes without requiring caller prebuilt matrix; repeated nodes yield exact singular result.

When to use Vandermonde Determinant 3 Calculator

Use this tool to compute oriented determinant of rows [1,x,x²] from three nodes without requiring caller prebuilt matrix; repeated nodes yield exact singular result.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Vandermonde Determinant 3 Calculator and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter nodes using the field table and calculation rules, then run the tool.
Read vandermondeDeterminant, distinctNodes. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
nodes	Nodes	list of number	Required

Enter exactly three finite nodes. Evaluate the oriented determinant for rows [1,x,x²] and separately report whether all three nodes are distinct. Repeated nodes validly return determinant zero.

determinant = (x2−x1)(x3−x1)(x3−x2)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
vandermondeDeterminant	Vandermonde determinant	number
distinctNodes	Distinct nodes	boolean

json

{
  "input": {
    "nodes": [
      1,
      2,
      4
    ]
  },
  "output": {
    "vandermondeDeterminant": 6,
    "distinctNodes": true
  }
}

Limitations and assumptions

Floating-point arithmetic can overflow or lose precision for extreme node scales. No arbitrary-size matrix is constructed.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Numeric lists are limited to 100,000 entries; the rules above may require smaller limits. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use Sylvester Resultant Calculator: Compute exact integer resultant by fraction-free Sylvester determinant for constant-first integer polynomials of degree ≤12; zero polynomial returns zero resultant. Nonzero constants have no common roots. For a complementary task, use 2x2 Characteristic Polynomial Calculator: Construct det(λI−A) in ascending coefficient order for real 2×2 input, not a generic trace/determinant alias; preserves full characteristic polynomial.

References

Vandermonde Determinant 3 Calculator — tool.io

Similar or alternative tools

Sylvester Resultant Calculator
Compute exact integer resultant by fraction-free Sylvester determinant for constant-first integer polynomials of degree ≤12; zero polynomial returns zero resultant. Nonzero constants have no common roots.
2x2 Characteristic Polynomial Calculator
Construct det(λI−A) in ascending coefficient order for real 2×2 input, not a generic trace/determinant alias; preserves full characteristic polynomial.