pascal

Pascal matrix

Syntax

P = pascal(n)

P = pascal(n,1)

P = pascal(n,2)

P = pascal(___,classname)

Description

P = pascal(n) returns a Pascal Matrix of order n. P is a symmetric positive definite matrix with integer entries taken from Pascal's triangle. The inverse of P has integer entries.

P = pascal(n,1) returns the lower triangular Cholesky factor (up to the signs of the columns) of the Pascal matrix. P is involutory, that is, it is its own inverse.

P = pascal(n,2) returns a transposed and permuted version of pascal(n,1). In this case, P is a cube root of the identity matrix.

P = pascal(___,classname) returns a matrix of class classname using any of the input argument combinations in previous syntaxes. classname can be "double" or "single".

Examples

Matrix from Pascal's Triangle

Compute the fourth-order Pascal matrix.

A = pascal(4)

A = 4×4

     1     1     1     1
     1     2     3     4
     1     3     6    10
     1     4    10    20

Compute the lower triangular Cholesky factor of the third-order Pascal matrix, and verify it is involutory.

A = pascal(3,1)

A = 3×3

     1     0     0
     1    -1     0
     1    -2     1

inv(A)

ans = 3×3

     1     0     0
     1    -1     0
     1    -2     1

Input Arguments

n — Matrix order

scalar, nonnegative integer

Matrix order, specified as a scalar, nonnegative integer.

Example: pascal(10)

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical | char

classname — Matrix class

"double" (default) | "single"

Matrix class, specified as either "double" or "single".

Example: pascal(10,'single')

Data Types: char

Output Arguments

P — Pascal matrix

matrix

Pascal matrix, returned as a matrix.

Limitations

• When the elements of a Pascal matrix are too large, exceeding the largest positive floating-point number, pascal returns Inf for those elements. This largest positive floating-point number is approximately 1.79e308 for double precision and approximately 3.40e38 for single precision. For example, this code returns Inf.

  P = pascal(516);
  P(end)

More About

Pascal Matrix

Pascal’s triangle is a triangle formed by rows of numbers. The first row has entry 1. Each succeeding row is formed by adding adjacent entries of the previous row, substituting a 0 where no adjacent entry exists. The pascal function forms Pascal matrix by selecting the portion of Pascal’s triangle that corresponds to the specified matrix dimensions, as outlined in the graphic. The matrix outlined corresponds to the MATLAB^® command pascal(4).

Extended Capabilities

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

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

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

Version History

Introduced before R2006a