invhilb

Inverse of Hilbert matrix

Syntax

H = invhilb(n)

H = invhilb(n,typename)

Description

H = invhilb(n) generates the exact inverse of the exact Hilbert matrix for n less than about 15. For larger n, the invhilb function generates an approximation to the inverse Hilbert matrix.

example

H = invhilb(n,typename) returns a matrix of type typename, which can be either "single" or "double".

Examples

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Inverse Hilbert Matrix

Open Live Script

Compute the fourth-order inverse Hilbert matrix.

invhilb(4)

ans = 4×4

          16        -120         240        -140
        -120        1200       -2700        1680
         240       -2700        6480       -4200
        -140        1680       -4200        2800

Input Arguments

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`n` — Matrix order
scalar, nonnegative integer

Matrix order, specified as a scalar, nonnegative integer.

Example: invhilb(10)

`typename` — Matrix data type
`"double"` (default) | `"single"`

Matrix data type, specified as either "double" or "single".

Example: invhilb(10,"single")

Limitations

The exact inverse of the exact Hilbert matrix is a matrix whose elements are large integers. As long as the order of the matrix n is less than 15, these integers can be represented as floating-point numbers without roundoff error.

Comparing invhilb(n) with inv(hilb(n)) involves the effects of two or three sets of roundoff errors:

Errors caused by representing hilb(n)
Errors in the matrix inversion process
Errors, if any, in representing invhilb(n)

The first of these roundoff errors involves representing fractions like 1/3 and 1/5 in floating-point representation and is the most significant.

References

[1] Forsythe, G. E. and C. B. Moler. Computer Solution of Linear Algebraic Systems. Englewood Cliffs, NJ: Prentice-Hall, 1967.

Extended Capabilities

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C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.

Thread-Based Environment
Run code in the background using MATLAB® `backgroundPool` or accelerate code with Parallel Computing Toolbox™ `ThreadPool`.

The invhilb function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

Version History

Introduced before R2006a