Hilbert Matrix Entry Calculator

Hilbertmatrixentry1/(i+j−1),actualonebasedintegerindices.

Description

The Hilbert matrix uses reciprocal sums of one-based row and column indices.

The Hilbert matrix uses reciprocal sums of one-based row and column indices.^[1]

This tool returns a specified entry without constructing or inverting the matrix.

When to use Hilbert Matrix Entry Calculator

This tool returns a specified entry without constructing or inverting the matrix.

How to use Hilbert Matrix Entry Calculator and how it works

Prepare one based row index, one based column index, using the input definitions and matching units below.
Enter the required fields and run Hilbert Matrix Entry Calculator.
Read result.value using the units, signs and ordering explained below.
Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.

Input field	Meaning	Value type	Requirement
actualOneBasedRowIndex	Hilbert-matrix row index beginning at 1.	number	Required; integer 1–1000000; finite; other constraints below
actualOneBasedColumnIndex	Hilbert-matrix column index beginning at 1.	number	Required; integer 1–1000000; finite; other constraints below

Require row and column indices as one-based integers 1–1000000. Return 1/(row+column-1) as a binary64 Hilbert-matrix entry. No matrix inversion or zero-based indexing is accepted.

entry=1/(oneBasedRow+oneBasedColumn-1)

Apply the input ordering, units and branch conditions stated above.

Output field	Meaning	Value type
result	Object containing the calculated or selected fields below.	object
result.value	Hilbert-matrix entry at the supplied one-based indices.	number

json

{
  "input": {
    "actualOneBasedRowIndex": 2,
    "actualOneBasedColumnIndex": 3
  },
  "output": {
    "result": {
      "value": 0.25
    }
  }
}

Limitations and assumptions

The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.

All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.

Alternative or Complementary methods

Use Subfactorial Calculator for a complementary operation: A subfactorial counts permutations with no fixed points.

Use Pascal Determinant Calculator for a complementary operation: The leading symmetric Pascal matrix factors into unit triangular matrices, giving determinant one.

References

Hilbert Matrix Entry Calculator — tool.io

Similar or alternative tools

Subfactorial Calculator
A subfactorial counts permutations with no fixed points.
Pascal Determinant Calculator
The leading symmetric Pascal matrix factors into unit triangular matrices, giving determinant one.