CauchyMatrix

CauchyMatrix[x,y]

represents the Cauchy matrix given by the generating vectors x and y as a structured array.

CauchyMatrix[x]

is equivalent to CauchyMatrix[x,x].

CauchyMatrix[cmat]

converts a Cauchy matrix cmat to a structured array.

Details and Options

  • Cauchy matrices, when represented as structured arrays, allow for efficient storage and more efficient operations, including Det, Inverse and LinearSolve.
  • Cauchy matrices occur in computations related to rational interpolation, conformal mappings, n-body simulations and the discretization of integral equations with singular kernels.
  • Given generating vectors x and y, the resulting Cauchy matrix has entries given by .
  • Operations that are accelerated for CauchyMatrix include:
  • Dettime
    Inversetime
    LinearSolvetime
  • For a CauchyMatrix sa, the following properties "prop" can be accessed as sa["prop"]:
  • "XVector"generating vector x
    "YVector"generating vector y
    "Properties"list of supported properties
    "Structure"type of structured array
    "StructuredData"internal data stored by the structured array
    "StructuredAlgorithms"list of functions with special methods for the structured array
    "Summary"summary information, represented as a Dataset
  • Normal[CauchyMatrix[x]] gives the Cauchy matrix as an ordinary matrix.
  • CauchyMatrix[…,TargetStructure->struct] returns the Cauchy matrix in the format specified by struct. Possible settings include:
  • Automaticautomatically choose the representation returned
    "Dense"represent the matrix as a dense matrix
    "Structured"represent the matrix as a structured array
    "Symmetric"represent the matrix as a symmetric matrix
  • CauchyMatrix[…,TargetStructureAutomatic] is equivalent to CauchyMatrix[…,TargetStructure"Structured"].

Examples

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Basic Examples  (2)

Construct a Cauchy matrix:

Show the elements:

Normal can convert a CauchyMatrix to its ordinary representation:

Construct a Cauchy matrix with symbolic entries:

Show the elements:

Get the determinant:

Scope  (6)

Generate a symmetric Cauchy matrix:

Show the elements:

Get the normal representation:

Generate a rectangular Cauchy matrix:

Show the elements:

Get the normal representation:

Represent a dense Cauchy matrix as a structured array:

Show the elements:

The structured representation typically uses much less memory:

CauchyMatrix objects include properties that give information about the array:

The "XVector" and "YVector" properties give the generating vectors of the Cauchy matrix:

The "Summary" property gives a brief summary of information about the array:

The "StructuredAlgorithms" property lists the functions that have structured algorithms:

When appropriate, structured algorithms return another CauchyMatrix object:

The transpose is also a CauchyMatrix:

The product of a Cauchy matrix and its transpose is no longer a Cauchy matrix:

Options  (1)

TargetStructure  (1)

Return the Cauchy matrix as a dense matrix:

Return the Cauchy matrix as a structured array:

Return the Cauchy matrix as a symmetric matrix:

Applications  (3)

Represent the Hilbert matrix as a CauchyMatrix:

Compare with HilbertMatrix:

Numerically compute the inverse for a large Hilbert matrix:

The result from the structured version is more accurate:

Use CauchyMatrix to compute the coefficients for an interpolating rational function with fixed poles:

Construct the rational function:

Visualize the interpolant in the complex plane:

Check that the rational interpolant does pass through all the given points:

Define the Parter matrix as a CauchyMatrix:

The Parter matrix is both a Cauchy matrix and a Toeplitz matrix:

The Parter matrix's dominant singular values are clustered around :

Properties & Relations  (2)

Express a Cauchy matrix as a product of diagonal matrices, Vandermonde matrices and their inverses:

Express a Cauchy matrix as a product of diagonal, Vandermonde and Hankel matrices:

Possible Issues  (3)

The literature sometimes uses a different definition of the Cauchy matrix, where the second generating vector is negated:

If any of the generating vectors have repeated entries, a Cauchy matrix cannot be constructed:

If a component of any one of the generating vectors is the negative of a component of the other generating vector, a Cauchy matrix cannot be constructed:

Wolfram Research (2022), CauchyMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/CauchyMatrix.html (updated 2023).

Text

Wolfram Research (2022), CauchyMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/CauchyMatrix.html (updated 2023).

CMS

Wolfram Language. 2022. "CauchyMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/CauchyMatrix.html.

APA

Wolfram Language. (2022). CauchyMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CauchyMatrix.html

BibTeX

@misc{reference.wolfram_2024_cauchymatrix, author="Wolfram Research", title="{CauchyMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/CauchyMatrix.html}", note=[Accessed: 25-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_cauchymatrix, organization={Wolfram Research}, title={CauchyMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/CauchyMatrix.html}, note=[Accessed: 25-April-2024 ]}