# FourierDSTMatrix

returns an n×n discrete sine transform matrix of type 2.

FourierDSTMatrix[n,m]

returns an n×n discrete sine transform matrix of type m.

# Details and Options

• Each entry Frs of the discrete sine transform matrix of type m is computed as:
•  1 DST-I 2 DST-II 3 DST-III 4 DST-IV
• The discrete sine transform matrices of types 1, 2, 3 and 4 have inverses of type 1, 3, 2 and 4, respectively. »
• Rows of the FourierDSTMatrix are basis sequences of the discrete sine transform.
• The result of FourierDSTMatrix[n].list is equivalent to FourierDST[list] when list has length n. However, the computation of FourierDST[list] is much faster and has less numerical error. »
• For types 1 and 4, the option TargetStructure is supported, which specifies the structure of the returned matrix. Possible settings for TargetStructure include:
•  Automatic automatically choose the representation returned "Dense" represent the matrix as a dense matrix "Hermitian" represent the matrix as a Hermitian matrix "Orthogonal" represent the matrix as an orthogonal matrix "Symmetric" represent the matrix as a symmetric matrix "Unitary" represent the matrix as a unitary matrix
• is equivalent to FourierDSTMatrix[,TargetStructure"Dense"].
• gives a matrix with entries of precision p.

# Examples

open allclose all

## Basic Examples(1)

A 4×4 DST matrix:

## Scope(1)

The discrete sine transform's basis sequences of length 128:

## Options(2)

### TargetStructure(1)

Return the DST matrix as a dense matrix:

Return the DST matrix as an orthogonal matrix:

Return the DST matrix as a symmetric matrix:

### WorkingPrecision(1)

Use machine precision:

Use arbitrary precision:

## Applications(1)

A tridiagonal Toeplitz matrix:

The matrix of its eigenvectors can be expressed as a diagonal rescaling of the discrete sine transform matrix of type 1:

Diagonalize the tridiagonal matrix:

## Properties & Relations(2)

A DST matrix multiplied by a vector is equivalent to the discrete sine transform of that vector:

FourierDST is much faster than the matrix-based computation:

A discrete sine transform matrix of type 1 is its own inverse:

A discrete sine transform matrix of type 3 is an inverse of the type 2 matrix:

A discrete sine transform matrix of type 4 is its own inverse:

Wolfram Research (2012), FourierDSTMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierDSTMatrix.html (updated 2023).

#### Text

Wolfram Research (2012), FourierDSTMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/FourierDSTMatrix.html (updated 2023).

#### CMS

Wolfram Language. 2012. "FourierDSTMatrix." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/FourierDSTMatrix.html.

#### APA

Wolfram Language. (2012). FourierDSTMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FourierDSTMatrix.html

#### BibTeX

@misc{reference.wolfram_2024_fourierdstmatrix, author="Wolfram Research", title="{FourierDSTMatrix}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/FourierDSTMatrix.html}", note=[Accessed: 30-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_fourierdstmatrix, organization={Wolfram Research}, title={FourierDSTMatrix}, year={2023}, url={https://reference.wolfram.com/language/ref/FourierDSTMatrix.html}, note=[Accessed: 30-May-2024 ]}