math

class Matrix

Matrix.

Public Functions

explicit Matrix(int num_row = 0, int num_column = 0)

Parameters:

• num_row – [in] Number of rows.

• num_column – [in] Number of columns.

Matrix(int num_row, int num_column, const std::vector<double> &vector)

Parameters:

• num_row – [in] Number of rows.

• num_column – [in] Number of columns.

• vector – [in] Diagonal elements.

Matrix(const Matrix &matrix)

Parameters:

matrix – [in] Matrix.

Matrix &operator=(const Matrix &matrix)

Parameters:

matrix – [in] Matrix.

inline int GetNumRow() const

Returns:

Number of rows.

inline int GetNumColumn() const

Returns:

Number of columns.

void Resize(int num_row, int num_column)

Resize matrix.

Parameters:

• num_row – [in] Number of rows.

• num_column – [in] Number of columns.

inline double *operator[](int row)

Parameters:

row – [in] Row.

inline const double *operator[](int row) const

Parameters:

row – [in] Row.

double &At(int row, int column)

Get element.

Parameters:

• row – [in] i.

• column – [in] j.

Returns:

(i, j)-th element.

const double &At(int row, int column) const

Get element.

Parameters:

• row – [in] i.

• column – [in] j.

Returns:

(i, j)-th element.

Matrix &operator+=(const Matrix &matrix)

Parameters:

matrix – [in] Addend.

Returns:

Sum.

Matrix &operator-=(const Matrix &matrix)

Parameters:

matrix – [in] Subtrahend.

Returns:

Difference.

Matrix &operator*=(const Matrix &matrix)

Parameters:

matrix – [in] Multiplicand.

Returns:

Product.

Matrix operator+(const Matrix &matrix) const

Parameters:

matrix – [in] Addend.

Returns:

Sum.

Matrix operator-(const Matrix &matrix) const

Parameters:

matrix – [in] Subtrahend.

Returns:

Difference.

Matrix operator*(const Matrix &matrix) const

Parameters:

matrix – [in] Multiplicand.

Returns:

Product.

Matrix operator-() const

Negate.

Returns:

Negated matrix.

void Fill(double value)

Overwrite all elements with a value.

Parameters:

value – [in] Value.

void FillDiagonal(double value)

Overwrite diagonal elements with a value.

Parameters:

value – [in] Diagonal value.

void Negate()

Negate all elements of matrix.

bool Transpose(Matrix *transposed_matrix) const

Transpose matrix.

Parameters:

transposed_matrix – [out] Transposed matrix.

Returns:

True on success, false on failure.

bool GetSubmatrix(int row_offset, int num_row_of_submatrix, int column_offset, int num_column_of_submatrix, Matrix *submatrix) const

Get submatrix.

Parameters:

• row_offset – [in] Offset of row.

• num_row_of_submatrix – [in] Number of rows of submatrix.

• column_offset – [in] Offset of column.

• num_column_of_submatrix – [in] Number of columns of submatrix.

• submatrix – [out] Submatrix.

Returns:

True on success, false on failure.

bool GetDeterminant(double *determinant) const

Compute determinant.

Parameters:

determinant – [out] Determinant.

Returns:

True on success, false on failure.

class Matrix2D

2D matrix.

Public Functions

Matrix2D()

Make 2D matrix.

Matrix2D(const Matrix2D &matrix)

Parameters:

matrix – [in] 2D matrix.

Matrix2D &operator=(const Matrix2D &matrix)

Parameters:

matrix – [in] 2D matrix.

inline double *operator[](int row)

Parameters:

row – [in] Row.

inline const double *operator[](int row) const

Parameters:

row – [in] Row.

double &At(int row, int column)

Get element.

Parameters:

• row – [in] i.

• column – [in] j.

Returns:

(i, j)-th element.

const double &At(int row, int column) const

Get element.

Parameters:

• row – [in] i.

• column – [in] j.

Returns:

(i, j)-th element.

void Fill(double value)

Overwrite all elements with a value.

Parameters:

value – [in] Value.

void FillDiagonal(double value)

Overwrite diagonal elements with a value.

Parameters:

value – [in] Diagonal value.

void Negate()

Negate all elements of matrix.

void Negate(const Matrix2D &matrix)

Negate all elements of matrix.

bool CrossTranspose(Matrix2D *transposed_matrix) const

Compute cross-transpose matrix.

Parameters:

transposed_matrix – [out] Cross-transposed matrix.

Returns:

True on success, false on failure.

bool Invert(Matrix2D *inverse_matrix) const

Compute inverse matrix.

Parameters:

inverse_matrix – [out] Inverse matrix.

Returns:

True on success, false on failure.

Public Static Functions

static bool Add(const Matrix2D &matrix, Matrix2D *output)

Compute sum.

Parameters:

• matrix – [in] Addend.

• output – [inout] Result.

Returns:

True on success, false on failure.

static bool Add(const Matrix2D &first_matrix, const Matrix2D &second_matrix, Matrix2D *output)

Compute sum.

Parameters:

• first_matrix – [in] Augend.

• second_matrix – [in] Addend.

• output – [out] Result.

Returns:

True on success, false on failure.

static bool Subtract(const Matrix2D &matrix, Matrix2D *output)

Compute difference.

Parameters:

• matrix – [in] Subtrahend.

• output – [inout] Result.

Returns:

True on success, false on failure.

static bool Subtract(const Matrix2D &first_matrix, const Matrix2D &second_matrix, Matrix2D *output)

Compute difference.

Parameters:

• first_matrix – [in] Minuend.

• second_matrix – [in] Subtrahend.

• output – [out] Result.

Returns:

True on success, false on failure.

static bool Multiply(const Matrix2D &matrix, const std::vector<double> &column_vector, std::vector<double> *output)

Compute Ax.

Parameters:

• matrix – [in] A.

• column_vector – [in] x.

• output – [out] Result.

Returns:

True on success, false on failure.

static bool Multiply(const Matrix2D &first_matrix, const Matrix2D &second_matrix, Matrix2D *output)

Compute AB.

Parameters:

• first_matrix – [in] A.

• second_matrix – [in] B.

• output – [out] Result.

Returns:

True on success, false on failure.

class SymmetricMatrix

Symmetric matrix.

Public Functions

explicit SymmetricMatrix(int num_dimension = 0)

Parameters:

num_dimension – [in] Size of matrix.

SymmetricMatrix(const SymmetricMatrix &matrix)

Parameters:

matrix – [in] Symmetric matrix.

SymmetricMatrix &operator=(const SymmetricMatrix &matrix)

Parameters:

matrix – [in] Symmetric matrix.

inline int GetNumDimension() const

Returns:

Number of dimensions.

void Resize(int num_dimension)

Resize matrix.

Parameters:

num_dimension – [in] Number of dimensions.

inline Row operator[](int row)

Parameters:

row – [in] Row.

inline const Row operator[](int row) const

Parameters:

row – [in] Row.

double &At(int row, int column)

Get element.

Parameters:

• row – [in] i.

• column – [in] j.

Returns:

(i, j)-th element.

const double &At(int row, int column) const

Get element.

Parameters:

• row – [in] i.

• column – [in] j.

Returns:

(i, j)-th element.

inline bool Empty() const

Check whether the matrix is empty.

Returns:

True if the matrix is empty.

void Fill(double value)

Overwrite all elements with a value.

Parameters:

value – [in] Value.

bool GetDiagonal(std::vector<double> *diagonal_elements) const

Get diagonal elements.

Parameters:

diagonal_elements – [out] Diagonal elements.

Returns:

True on success, false on failure.

bool SetDiagonal(const std::vector<double> &diagonal_elements) const

Set diagonal elements.

Parameters:

diagonal_elements – [in] Diagonal elements.

Returns:

True on success, false on failure.

bool CholeskyDecomposition(SymmetricMatrix *lower_triangular_matrix, std::vector<double> *diagonal_elements) const

Perform Cholesky decomposition.

Parameters:

• lower_triangular_matrix – [out] Lower triangular matrix.

• diagonal_elements – [out] Diagonal elements.

Returns:

True on success, false on failure.

bool Invert(SymmetricMatrix *inverse_matrix) const

Compute inverse matrix.

Parameters:

inverse_matrix – [out] Inverse matrix.

Returns:

True on success, false on failure.

class Row

A row of symmetric matrix.a

Public Functions

inline Row(const SymmetricMatrix &matrix, int row)

Parameters:

• matrix – [in] Symmetric matrix.

• row – [in] Row index.

double &operator[](int column)

Parameters:

column – [in] Column index.

Returns:

Element.

const double &operator[](int column) const

Parameters:

column – [in] Column index.

Returns:

Element.

class SymmetricSystemSolver

Solve the following symmetric system:

\[ \boldsymbol{A} \boldsymbol{x} = \boldsymbol{b}, \]

where \(\boldsymbol{A}\) is a symmetric matrix.

The inputs are \(\boldsymbol{A}\) and \(M\)-th order constant vector:

\[ \begin{array}{cccc} b(0), & b(1), & \ldots, & b(M). \end{array} \]

The outputs are the unknown coefficients

\[ \begin{array}{cccc} x(0), & x(1), & \ldots, & x(M). \end{array} \]

Public Functions

explicit SymmetricSystemSolver(int num_order)

Parameters:

num_order – [in] Order of vector, \(M\).

inline int GetNumOrder() const

Returns:

Order of vector.

inline bool IsValid() const

Returns:

True if this object is valid.

bool Run(const SymmetricMatrix &coefficient_matrix, const std::vector<double> &constant_vector, std::vector<double> *solution_vector, SymmetricSystemSolver::Buffer *buffer) const

Parameters:

• coefficient_matrix – [in] \((M+1, M+1)\) matrix \(\boldsymbol{A}\).

• constant_vector – [in] \(M\)-th order vector \(\boldsymbol{b}\).

• solution_vector – [out] \(M\)-th order vector \(\boldsymbol{x}\).

• buffer – [out] Buffer.

Returns:

True on success, false on failure.

class Buffer

Buffer for SymmetricSystemSolver class.

class ToeplitzPlusHankelSystemSolver

Solve the following Toeplitz-plus-Hankel system:

\[ (\boldsymbol{T} + \boldsymbol{H}) \boldsymbol{x} = \boldsymbol{b}, \]

where

\[\begin{split}\begin{eqnarray} \boldsymbol{T} &=& \left[ \begin{array}{ccccc} t(0) & t(-1) & t(-2) & \cdots & t(-M) \\ t(1) & t(0) & t(-1) & \cdots & t(-M+1) \\ t(2) & t(1) & t(0) & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & t(-1) \\ t(M) & t(M-1) & \cdots & t(1) & t(0) \end{array} \right], \\ \boldsymbol{H} &=& \left[ \begin{array}{ccccc} h(0) & h(1) & h(2) & \cdots & h(M) \\ h(1) & h(2) & h(3) & \cdots & h(M+1) \\ h(2) & t(3) & h(4) & \ddots & \vdots \\ \vdots & \vdots & \ddots & \ddots & h(2M-1) \\ h(M) & h(M+1) & \cdots & h(2M-1)& h(2M) \end{array} \right], \\ \boldsymbol{b} &=& \left[ \begin{array}{cccc} b(0) & b(1) & \cdots & b(M) \end{array} \right]^\mathsf{T}. \\ \end{eqnarray}\end{split}\]

The inputs are \((2M+1)\) Toeplitz coefficient vector:

\[ \begin{array}{cccc} t(-M), & t(-M+1), & \ldots, & t(M), \end{array} \]

\((2M+1)\) Hankel coefficient vector:

\[ \begin{array}{cccc} h(0), & h(1), & \ldots, & h(2M), \end{array} \]

and \((M+1)\) constant vector:

\[ \begin{array}{cccc} b(0), & b(1), & \ldots, & b(M). \end{array} \]

The outputs are the unknown coefficients

\[ \begin{array}{cccc} x(0), & x(1), & \ldots, & x(M). \end{array} \]

[1] G. Merchant and T. Parks, “Efficient solution of a Toeplitz-plus-Hankel coefficient matrix system of equations,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 30, no. 1, pp. 40-44, 1982.

Public Functions

explicit ToeplitzPlusHankelSystemSolver(int num_order, bool coefficients_modification = true)

Parameters:

• num_order – [in] Order of vector, \(M\).

• coefficients_modification – [in] If true, perform coefficients modification.

inline int GetNumOrder() const

Returns:

Order of vector.

inline bool GetCoefficientsModificationFlag()

Returns:

Ture if coefficients modification is enabled.

inline bool IsValid() const

Returns:

True if this object is valid.

bool Run(const std::vector<double> &toeplitz_coefficient_vector, const std::vector<double> &hankel_coefficient_vector, const std::vector<double> &constant_vector, std::vector<double> *solution_vector, ToeplitzPlusHankelSystemSolver::Buffer *buffer) const

Parameters:

• toeplitz_coefficient_vector – [in] \(2M\)-th order coefficient vector of Toeplitz matrix \(\boldsymbol{T}\).

• hankel_coefficient_vector – [in] \(2M\)-th order coefficient vector of Hankel matrix \(\boldsymbol{H}\).

• constant_vector – [in] \(M\)-th order constant vector \(\boldsymbol{b}\).

• solution_vector – [out] \(M\)-th order solution vector \(\boldsymbol{x}\).

• buffer – [out] Buffer.

Returns:

True on success, false on failure.

class Buffer

Buffer for ToeplitzPlusHankelSystemSolver class.

class VandermondeSystemSolver

Solve the following Vandermonde system:

\[\begin{split} \left[ \begin{array}{cccc} 1 & 1 & \cdots & 1 \\ x(0) & x(1) & \cdots & x(M) \\ x^2(0) & x^2(1) & \cdots & x^2(M) \\ \vdots & \vdots & \ddots & \vdots \\ x^M(0) & x^M(1) & \cdots & x^M(M) \end{array} \right] \left[ \begin{array}{c} w(0) \\ w(1) \\ w(2) \\ \vdots \\ w(M) \end{array} \right] = \left[ \begin{array}{c} q(0) \\ q(1) \\ q(2) \\ \vdots \\ q(M) \end{array} \right]. \end{split}\]

The inputs are

\[ \begin{array}{cccc} x(0), & x(1), & \ldots, & x(M), \end{array} \]

and

\[ \begin{array}{cccc} q(0), & q(1), & \ldots, & q(M). \end{array} \]

The outputs are the unknown coefficients

\[ \begin{array}{cccc} w(0), & w(1), & \ldots, & w(M). \end{array} \]

[1] W. H. Press, et al., “Numerical recipes in C: The art of scientific computing,” Cambridge University Press, pp. 90-92, 1992.

Public Functions

explicit VandermondeSystemSolver(int num_order)

Parameters:

num_order – [in] Order of vector, \(M\).

inline int GetNumOrder() const

Returns:

Order of vector.

inline bool IsValid() const

Returns:

True if this object is valid.

bool Run(const std::vector<double> &coefficient_vector, const std::vector<double> &constant_vector, std::vector<double> *solution_vector, VandermondeSystemSolver::Buffer *buffer) const

Parameters:

• coefficient_vector – [in] \(M\)-th order vector \(\boldsymbol{x}\).

• constant_vector – [in] \(M\)-th order vector \(\boldsymbol{q}\).

• solution_vector – [out] \(M\)-th order vector \(\boldsymbol{w}\).

• buffer – [out] Buffer.

Returns:

True on success, false on failure.

class Buffer

Buffer for VandermondeSystemSolver class.