Classes
class	Complex
	Class for complex numbers. More...

struct	LUP
	Struct for LUP decomposition. More...

struct	QR
	Struct for QR decomposition. More...

struct	Cholesky
	Struct for Cholesky decomposition. More...

struct	Eigendecomposition
	Struct for Eigendecomposition. More...

struct	SVD
	Struct for SVD decomposition. More...

struct	Hessenberg
	Struct for Hessenberg decomposition. More...

struct	Schur
	Struct for Schur decomposition. More...

struct	Eigenpair
	Struct for eigenpairs. More...

class	Matrix
	Class for matrices. More...

Typedefs
using	Complexi = Complex< int >

using	Complexf = Complex< float >

using	Complexd = Complex< double >

template<typename T , size_t N, unsigned int Flags = 0>
using	SquareMatrix = Matrix< T, N, N, Flags >

using	Matrix2f = Matrix< float, 2, 2 >

using	Matrix2x2f = Matrix< float, 2, 2 >

using	Matrix2x3f = Matrix< float, 2, 3 >

using	Matrix2x4f = Matrix< float, 2, 4 >

using	Matrix3f = Matrix< float, 3, 3 >

using	Matrix3x2f = Matrix< float, 3, 2 >

using	Matrix3x3f = Matrix< float, 3, 3 >

using	Matrix3x4f = Matrix< float, 3, 4 >

using	Matrix4f = Matrix< float, 4, 4 >

using	Matrix4x2f = Matrix< float, 4, 2 >

using	Matrix4x3f = Matrix< float, 4, 3 >

using	Matrix4x4f = Matrix< float, 4, 4 >

using	MatrixXf = Matrix< float, Dynamic, Dynamic >

using	Matrix2d = Matrix< double, 2, 2 >

using	Matrix2x2d = Matrix< double, 2, 2 >

using	Matrix2x3d = Matrix< double, 2, 3 >

using	Matrix2x4d = Matrix< double, 2, 4 >

using	Matrix3d = Matrix< double, 3, 3 >

using	Matrix3x2d = Matrix< double, 3, 2 >

using	Matrix3x3d = Matrix< double, 3, 3 >

using	Matrix3x4d = Matrix< double, 3, 4 >

using	Matrix4d = Matrix< double, 4, 4 >

using	Matrix4x2d = Matrix< double, 4, 2 >

using	Matrix4x3d = Matrix< double, 4, 3 >

using	Matrix4x4d = Matrix< double, 4, 4 >

using	MatrixXd = Matrix< double, Dynamic, Dynamic >

using	Matrix2ld = Matrix< long double, 2, 2 >

using	Matrix2x2ld = Matrix< long double, 2, 2 >

using	Matrix2x3ld = Matrix< long double, 2, 3 >

using	Matrix2x4ld = Matrix< long double, 2, 4 >

using	Matrix3ld = Matrix< long double, 3, 3 >

using	Matrix3x2ld = Matrix< long double, 3, 2 >

using	Matrix3x3ld = Matrix< long double, 3, 3 >

using	Matrix3x4ld = Matrix< long double, 3, 4 >

using	Matrix4ld = Matrix< long double, 4, 4 >

using	Matrix4x2ld = Matrix< long double, 4, 2 >

using	Matrix4x3ld = Matrix< long double, 4, 3 >

using	Matrix4x4ld = Matrix< long double, 4, 4 >

using	MatrixXld = Matrix< long double, Dynamic, Dynamic >

template<typename T , size_t N>
using	Vector = Matrix< T, N, 1 >

using	Vector2f = Vector< float, 2 >

using	Vector3f = Vector< float, 3 >

using	Vector4f = Vector< float, 4 >

using	VectorXf = Vector< float, Dynamic >

using	Vector2d = Vector< double, 2 >

using	Vector3d = Vector< double, 3 >

using	Vector4d = Vector< double, 4 >

using	VectorXd = Vector< double, Dynamic >

using	Vector2ld = Vector< long double, 2 >

using	Vector3ld = Vector< long double, 3 >

using	Vector4ld = Vector< long double, 4 >

using	VectorXld = Vector< long double, Dynamic >

template<typename T , size_t N>
using	RowVector = Matrix< T, 1, N >

using	RowVector2f = RowVector< float, 2 >

using	RowVector3f = RowVector< float, 3 >

using	RowVector4f = RowVector< float, 4 >

using	RowVectorXf = RowVector< float, Dynamic >

using	RowVector2d = RowVector< double, 2 >

using	RowVector3d = RowVector< double, 3 >

using	RowVector4d = RowVector< double, 4 >

using	RowVectorXd = RowVector< double, Dynamic >

using	RowVector2ld = RowVector< long double, 2 >

using	RowVector3ld = RowVector< long double, 3 >

using	RowVector4ld = RowVector< long double, 4 >

using	RowVectorXld = RowVector< long double, Dynamic >

Enumerations
enum	SVDType { FULL_SVD , THIN_SVD }

Functions
template<size_t P, size_t Q, typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, P, Q, Flags >	SubMatrix (Matrix< T, M, N, Flags > A, size_t i=0, size_t j=0)
	Returns the PxQ submatrix located at (i,j) If i+P>M or j+Q > N, an exception is raised. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, Dynamic, Dynamic, Flags >	SubMatrix (Matrix< T, M, N, Flags > A, size_t nrows, size_t ncols, size_t i=0, size_t j=0)
	Returns the nrowsxncols submatrix located at (i,j) If i+nrows>M or j+ncols > N, an exception is raised. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T,(M==Dynamic?Dynamic:M-1), N, Flags >	RemoveRow (Matrix< T, M, N, Flags > A, size_t i)
	Returns the (M-1)xN submatrix formed by removing the ith row. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M,(N==Dynamic?Dynamic:N-1), Flags >	RemoveColumn (Matrix< T, M, N, Flags > A, size_t i)
	Returns the Mx(N-1) submatrix formed by removing the ith column. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T,(M==Dynamic?Dynamic:M-1),(N==Dynamic?Dynamic:N-1), Flags >	RemoveRowAndColumn (Matrix< T, M, N, Flags > A, size_t i, size_t j)
	Returns the (M-1)x(N-1) submatrix formed by removing the ith row and jth column. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, N, M, Flags >	Transpose (const Matrix< T, M, N, Flags > &A)
	Returns the NxM matrix $B=A^\top$ defined by $b_{ij}=a_{ji}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, N, M, Flags >	ConjugateTranspose (Matrix< T, M, N, Flags > A)
	Returns the NxM matrix $B=A^*$ defined by $b_{ij}=\overline{a_{ji}}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	RREF (Matrix< T, M, N, Flags > A)
	Computes A's reduced row echelon form. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Complex< T >	Trace (const Matrix< T, M, N, Flags > &A)
	Computes the trace of a square matrix $\sum_{i=0}^{N-1}a_{ii}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Complex< T >	Determinant (const Matrix< T, M, N, Flags > &A)
	Computes the determinant of a square matrix defined by $\sum_{i=0}^{N-1}(-1)^ia_{0i}\det(A_{0i})$ where $A_{0i}$ is the (N-1)x(N-1) matrix obtained by deleting the 0th row and ith column. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Complex< T >	Minor (const Matrix< T, M, N, Flags > &A, size_t i, size_t j)
	Computes the (i,j)-minor of a square matrix defined by $\det(A_{ij})$ where $A_{ij}$ is the (N-1)x(N-1) matrix obtained by deleting the ith row and jth column. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Complex< T >	Cofactor (const Matrix< T, M, N, Flags > &A, size_t i, size_t j)
	Computes the (i,j)-cofactor of a square matrix defined by $(-1)^{i+j}\det(A_{ij})$ where $A_{ij}$ is the (N-1)x(N-1) matrix obtained by deleting the ith row and jth column. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Adjugate (const Matrix< T, M, N, Flags > &A)
	Computes the NxN adjugate B of a square matrix A defined by $b_{ij}=c_{ji}$ where $c_{ij}$ represents the (i,j)-cofactor of A. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Inverse (const Matrix< T, M, N, Flags > &A)
	Computes the NxN inverse $A^{-1}=\frac{1}{\det A}Adj$ of a square matrix A where Adj is the adjugate of A. More...

template<typename T >
bool	IsReal (const Complex< T > &z)
	Checks if a complex number $z=a+bi$ is real. More...

template<typename T >
T	Abs (T x)
	Takes a real number $x$ and computes $\|x\|$. More...

template<typename T >
T	Abs (const Complex< T > &z)
	Takes a complex number $z=a+bi$ and computes $\|z\|=\sqrt{a^2+b^2}$. More...

template<typename T >
T	Sign (T x)
	Takes a real number $x$ and computes $sgn(x)$. More...

template<typename T >
Complex< T >	Sign (const Complex< T > &z)
	Takes a complex number $z$ and computes $sgn(z)$. More...

template<typename T >
T	Arg (const Complex< T > &z)
	Takes a complex number $z=a+bi$ and computes $arg(z)=atan2(b,a)$. More...

template<typename T >
Complex< T >	Conjugate (const Complex< T > &z)
	Takes a complex number $z=a+bi$ and computes it's conjugate $\bar{z}=a-bi$. More...

template<typename T >
Complex< T >	Sqrt (T x)
	Computes the square root of a real number $x$. More...

template<typename T >
Complex< T >	Sqrt (const Complex< T > &z)
	Takes a complex number $z=a+bi$ and computes it's principal square root $\sqrt{z}=\gamma+\delta i$, where $\gamma=\sqrt{\frac{a+\|z\|}{2}}$ and $\delta=(sgn b)\sqrt{\frac{-a+\|z\|}{2}}$. More...

template<typename T >
T	Exp (T x)
	Takes a real number $x$ and computes $e^x$. More...

template<typename T >
Complex< T >	Exp (const Complex< T > &z)
	Takes a complex number $z$ and computes $e^z$. More...

template<typename T >
T	Log (T x)
	Computes the natural log of a real number. More...

template<typename T >
Complex< T >	Log (const Complex< T > &z)
	Computes the principal natural log of a complex number $z$. More...

template<typename T >
T	Log (T x, T base)
	Computes the $\log_{base}x$. More...

template<typename T >
Complex< T >	Log (const Complex< T > &z, const Complex< T > &base)
	Computes the $\log_{base}z$. More...

template<typename T >
Complex< T >	Log (const Complex< T > &z, T base)
	Computes the $\log_{base}z$. More...

template<typename T >
Complex< T >	Log (T x, const Complex< T > &base)
	Computes the $\log_{base}x$. More...

template<typename T >
T	Pow (T x, T y)
	Computes the real number $x^y$. More...

template<typename T >
Complex< T >	Pow (const Complex< T > &z, const Complex< T > &w)
	Computes the complex number $z^w$. More...

template<typename T >
T	Mod (T x, T y)
	Computes the floating-point modulus $x\bmod y$. More...

template<typename T >
Complex< T >	Mod (const Complex< T > &z, T y)
	Returns the complex number $(a\bmod y)+(b\bmod y)i$ where $z=a+bi$. More...

template<typename T >
Complex< T >	Mod (const Complex< T > &z, const Complex< T > &w)
	Returns the complex number $(a\bmod c)+(b\bmod d)i$ where $z=a+bi$ and $w=b+di$. More...

template<typename T >
T	Sin (T x)
	Returns the real number $\sin x$. More...

template<typename T >
Complex< T >	Sin (const Complex< T > &z)
	Returns the complex number $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$. More...

template<typename T >
T	Cos (T x)
	Returns the real number $\cos x$. More...

template<typename T >
Complex< T >	Cos (const Complex< T > &z)
	Returns the complex number $\cos z=\frac{e^{iz}+e^{-iz}}{2}$. More...

template<typename T >
T	Tan (T x)
	Returns the real number $\tan x$. More...

template<typename T >
Complex< T >	Tan (const Complex< T > &z)
	Returns the complex number $\tan z=\frac{1}{i}\left(\frac{e^{2iz}-1}{e^{2iz}+1}\right)$. More...

template<typename T >
T	Csc (T x)
	Returns the real number $\csc x=1/\sin x$. More...

template<typename T >
Complex< T >	Csc (const Complex< T > &z)
	Returns the complex number $\csc z=\frac{2i}{e^{iz}-e^{-iz}}$. More...

template<typename T >
T	Sec (T x)
	Returns the real number $\sec x=1/\cos x$. More...

template<typename T >
Complex< T >	Sec (const Complex< T > &z)
	Returns the complex number $\sec z=\frac{2}{e^{iz}+e^{-iz}}$. More...

template<typename T >
T	Cot (T x)
	Returns the real number $\cot x=1/\tan x$. More...

template<typename T >
Complex< T >	Cot (const Complex< T > &z)
	Returns the complex number $\cot z=i\left(\frac{e^{2iz}+1}{e^{2iz}-1}\right)$. More...

template<typename T >
T	ASin (T x)
	Returns the real number $\sin^{-1}x$. More...

template<typename T >
Complex< T >	ASin (const Complex< T > &z)
	Returns the complex number $\sin^{-1}z=\frac{1}{i}\log\left(iz+\|1-z^2\|^{1/2}e^{\frac{i}{2}arg(1-z^2)}\right)$. More...

template<typename T >
T	ACos (T x)
	Returns the real number $\cos^{-1}x$. More...

template<typename T >
Complex< T >	ACos (const Complex< T > &z)
	Returns the complex number $\cos^{-1}z=\frac{1}{i}\log\left(z+i\|1-z^2\|^{1/2}e^{\frac{i}{2}arg(1-z^2)}\right)$. More...

template<typename T >
T	ATan (T x)
	Returns the real number $\tan^{-1}x$. More...

template<typename T >
Complex< T >	ATan (const Complex< T > &z)
	Returns the complex number $\tan^{-1}z=\frac{1}{2i}\log\left(\frac{i-z}{i+z}\right)$. More...

template<typename T >
T	ACsc (T x)
	Returns the real number $\csc^{-1}x=\sin^{-1}(1/x)$. More...

template<typename T >
Complex< T >	ACsc (const Complex< T > &z)
	Returns the complex number $\csc^{-1}z=\sin^{-1}(1/z)$. More...

template<typename T >
T	ASec (T x)
	Returns the real number $\sec^{-1}x=\cos^{-1}(1/x)$. More...

template<typename T >
Complex< T >	ASec (const Complex< T > &z)
	Returns the complex number $\sec^{-1}z=\cos^{-1}(1/z)$. More...

template<typename T >
T	ACot (T x)
	Returns the real number $\cot^{-1}x=\tan^{-1}(1/x)$. More...

template<typename T >
Complex< T >	ACot (const Complex< T > &z)
	Returns the complex number $\cot^{-1}z=\frac{1}{2i}\log\left(\frac{z+i}{z-i}\right)$. More...

template<typename T >
T	Sinh (T x)
	Returns the real number $\sinh x$. More...

template<typename T >
Complex< T >	Sinh (const Complex< T > &z)
	Returns the complex number $\sinh z=\frac{e^z-e^{-z}}{2}$. More...

template<typename T >
T	Cosh (T x)
	Returns the real number $\cosh x$. More...

template<typename T >
Complex< T >	Cosh (const Complex< T > &z)
	Returns the complex number $\cosh z=\frac{e^z+e^{-z}}{2}$. More...

template<typename T >
T	Tanh (T x)
	Returns the real number $\tanh x$. More...

template<typename T >
Complex< T >	Tanh (const Complex< T > &z)
	Returns the complex number $\tanh z=\frac{e^z-e^{-z}}{e^z+e^{-z}}$. More...

template<typename T >
T	Csch (T x)
	Returns the real number $csch x = 1/\sinh x$. More...

template<typename T >
Complex< T >	Csch (const Complex< T > &z)
	Returns the complex number $csch z=\frac{2}{e^z-e^{-z}}$. More...

template<typename T >
T	Sech (T x)
	Returns the real number $sech x = 1/\cosh x$. More...

template<typename T >
Complex< T >	Sech (const Complex< T > &z)
	Returns the complex number $sech z=\frac{2}{e^z+e^{-z}}$. More...

template<typename T >
T	Coth (T x)
	Returns the real number $\coth x = 1/\tanh x$. More...

template<typename T >
Complex< T >	Coth (const Complex< T > &z)
	Returns the complex number $\coth z=\frac{e^z+e^{-z}}{e^z-e^{-z}}$. More...

template<typename T >
T	ASinh (T x)
	Returns the real number $\sinh^{-1} x$. More...

template<typename T >
Complex< T >	ASinh (const Complex< T > &z)
	Returns the complex number $\sinh^{-1} z=\log\left(z+\|1+z^2\|^{1/2}e^{\frac{i}{2}arg(1+z^2)}\right)$. More...

template<typename T >
T	ACosh (T x)
	Returns the real number $\cosh^{-1} x$. More...

template<typename T >
Complex< T >	ACosh (const Complex< T > &z)
	Returns the complex number $\cosh^{-1} z=\log\left(z+\|z^2-1\|^{1/2}e^{\frac{i}{2}arg(z^2-1)}\right)$. More...

template<typename T >
T	ATanh (T x)
	Returns the real number $\tanh^{-1} x$. More...

template<typename T >
Complex< T >	ATanh (const Complex< T > &z)
	Returns the complex number $\tanh^{-1} z=\frac{1}{2}\log\left(\frac{1+z}{1-z}\right)$. More...

template<typename T >
T	ACsch (T x)
	Returns the real number $csch^{-1} x=\sinh^{-1}(1/x)$. More...

template<typename T >
Complex< T >	ACsch (const Complex< T > &z)
	Returns the complex number $csch^{-1} z=\sinh^{-1}(1/z)$. More...

template<typename T >
T	ASech (T x)
	Returns the real number $sech^{-1} x=\cosh^{-1}(1/x)$. More...

template<typename T >
Complex< T >	ASech (const Complex< T > &z)
	Returns the complex number $sech^{-1} z=\cosh^{-1}(1/z)$. More...

template<typename T >
T	ACoth (T x)
	Returns the real number $\coth^{-1} x=\tanh^{-1}(1/x)$. More...

template<typename T >
Complex< T >	ACoth (const Complex< T > &z)
	Returns the complex number $\coth^{-1} z=\frac{1}{2}\log\left(\frac{z+1}{z-1}\right)$. More...

template<typename T >
T	Ceil (T x)
	Returns the real number $\lceil x\rceil$. More...

template<typename T >
Complex< T >	Ceil (const Complex< T > &z)
	Returns the complex number $\lceil a\rceil+\lceil b\rceil i$ where $z=a+bi$. More...

template<typename T >
T	Floor (T x)
	Returns the real number $\lfloor x\rfloor$. More...

template<typename T >
Complex< T >	Floor (const Complex< T > &z)
	Returns the complex number $\lfloor a\rfloor+\lfloor b\rfloor i$ where $z=a+bi$. More...

template<typename T >
T	Round (T x)
	Rounds $x$ to the nearest integer. More...

template<typename T >
Complex< T >	Round (const Complex< T > &z)
	Rounds the real and imaginary parts of $z$ to the nearest integer. More...

template<typename T , size_t N, unsigned int Flags = 0>
SquareMatrix< T, N, Flags >	Identity ()
	Creates the NxN Identity matrix. More...

template<typename T , unsigned int Flags = 0>
SquareMatrix< T, Dynamic, Flags >	Identity (size_t n)
	Dynamically creates the nxn Identity matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags = 0>
Matrix< T, M, N, Flags >	Zero ()
	Creates the MxN all zeros matrix. More...

template<typename T , unsigned int Flags = 0>
Matrix< T, Dynamic, Dynamic, Flags >	Zero (size_t nrows, size_t ncols)
	Dynamically creates the nrowsxncols all zeros matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags = 0>
Matrix< T, M, N, Flags >	One ()
	Creates the MxN all ones matrix. More...

template<typename T , unsigned int Flags = 0>
Matrix< T, Dynamic, Dynamic, Flags >	One (size_t nrows, size_t ncols)
	Dynamically creates the nrowsxncols all ones matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags = 0>
Matrix< T, M, N, Flags >	Constant (Complex< T > z)
	Creates an MxN matrix with every entry set to z. More...

template<typename T , unsigned int Flags = 0>
Matrix< T, Dynamic, Dynamic, Flags >	Constant (size_t nrows, size_t ncols, Complex< T > z)
	Dynamically creates an nrowsxncols matrix with every entry set to v. More...

template<typename T , size_t M, size_t N, unsigned int Flags = 0>
std::enable_if<(M >0 &&N >0), Matrix< T, M, N, Flags > >::type	Basis (size_t i, size_t j)
	Creates an MxN basis matrix $A$ where all entries are set to 0 except $a_{ij}=1$. More...

template<typename T , unsigned int Flags = 0>
Matrix< T, Dynamic, Dynamic, Flags >	Basis (size_t nrows, size_t ncols, size_t i, size_t j)
	Dynamically creates an nrowsxncols basis matrix $A$ where all entries are set to 0 except $a_{ij}=1$. More...

void	SeedRandom (unsigned int seed)
	Seeds the random number generator used in Random. More...

template<typename T , size_t M, size_t N, unsigned int Flags = 0>
Matrix< T, M, N, Flags >	Random (Complex< T > min=Complex< T >(0.0), Complex< T > max=Complex< T >(1.0))
	Creates an MxN random matrix $A$ where all entries are randomly determined between min and max. More...

template<typename T , unsigned int Flags = 0>
Matrix< T, Dynamic, Dynamic, Flags >	Random (size_t nrows, size_t ncols, Complex< T > min=Complex< T >(0.0), Complex< T > max=Complex< T >(1.0))
	Dynamically creates an nrowsxncols random matrix $A$ where all entries are randomly determined between min and max. More...

template<typename T , size_t N, unsigned int Flags = 0>
SquareMatrix< T, N, Flags >	Householder (Vector< T, N > v)
	Creates an NxN Householder transformation $H$ defined by $H = I - 2vv^*.$. More...

template<typename T , size_t N, unsigned int Flags = 0>
SquareMatrix< T, N, Flags >	Householder (const Vector< T, N > &x, size_t k)
	Creates an NxN Householder transformation $H$ such that $Hx$ eliminates the last k elements. More...

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2>
std::enable_if<(M1==M2\|\|M1==Dynamic\|\|M2==Dynamic), Matrix< T, M1,(N1==Dynamic\|\|N2==Dynamic?Dynamic:N1+N2), Flags1 > >::type	Augmented (const Matrix< T, M1, N1, Flags1 > &left, const Matrix< T, M2, N2, Flags2 > &right)
	Creates the M1x(N1+N2) augmented matrix. More...

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2>
std::enable_if<(N1==N2\|\|N1==Dynamic\|\|N2==Dynamic), Matrix< T,(M1==Dynamic\|\|M2==Dynamic?Dynamic:M1+M2), N1, Flags1 > >::type	RowAugmented (const Matrix< T, M1, N1, Flags1 > &top, const Matrix< T, M2, N2, Flags2 > &bottom)
	Creates the (M1+M2)xN1 row augmented matrix. More...

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2, size_t M3, size_t N3, unsigned int Flags3, size_t M4, size_t N4, unsigned int Flags4>
std::enable_if<((M1+M3==M2+M4\|\|M1==Dynamic\|\|M2==Dynamic\|\|M3==Dynamic\|\|M4==Dynamic)&&(N1+N2==N3+N4\|\|N1==Dynamic\|\|N2==Dynamic\|\|N3==Dynamic\|\|N4==Dynamic)), Matrix< T,(M1==Dynamic\|\|M2==Dynamic\|\|M3==Dynamic\|\|M4==Dynamic?Dynamic:M1+M3),(N1==Dynamic\|\|N2==Dynamic\|\|N3==Dynamic\|\|N4==Dynamic?Dynamic:N1+N2), Flags1 > >::type	Block (const Matrix< T, M1, N1, Flags1 > &tl, const Matrix< T, M2, N2, Flags2 > &tr, const Matrix< T, M3, N3, Flags3 > &bl, const Matrix< T, M4, N4, Flags4 > &br)
	Creates the (M1+M3)x(N1+N2) block matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q>
std::enable_if<((M==P\|\|M==Dynamic\|\|P==Dynamic)&&(N==Q\|\|N==Dynamic\|\|Q==Dynamic)), Matrix< T,(M==Dynamic?Dynamic:M+1),(N==Dynamic?Dynamic:N+1), Flags > >::type	Block (const Matrix< T, M, N, Flags > &tl, const Vector< T, P > &tr, const RowVector< T, Q > &bl, Complex< T > br)
	Creates the (M+1)x(N+1) block matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q>
std::enable_if<((M==P\|\|M==Dynamic\|\|P==Dynamic)&&(N==Q\|\|N==Dynamic\|\|Q==Dynamic)), Matrix< T,(M==Dynamic?Dynamic:M+1),(N==Dynamic?Dynamic:N+1), Flags > >::type	Block (Complex< T > tl, const RowVector< T, Q > &tr, const Vector< T, P > &bl, const Matrix< T, M, N, Flags > &br)
	Creates the (M+1)x(N+1) block matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags1, size_t P, size_t Q, unsigned int Flags2>
Matrix< T, M P, Q N, Flags1 >	Kronecker (const Matrix< T, M, N, Flags1 > &A, const Matrix< T, P, Q, Flags2 > &B)
	Computes the (MP)x(NQ) block matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags1, size_t P, size_t Q, unsigned int Flags2>
SquareMatrix< T, N *Q, Flags1 >	KroneckerSum (const Matrix< T, M, N, Flags1 > &A, const Matrix< T, P, Q, Flags2 > &B)
	Computes the (NQ)x(NQ) Kronecker sum defined by. More...

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2>
Matrix< T,(M1==Dynamic\|\|M2==Dynamic?Dynamic:M1+M2),(N1==Dynamic\|\|N2==Dynamic?Dynamic:N1+N2), Flags1 >	Diag (const Matrix< T, M1, N1, Flags1 > &A, const Matrix< T, M2, N2, Flags2 > &B)
	Computes the (M1+M2)x(N1+N2) block diagonal matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T,(M==Dynamic?Dynamic:M+1),(N==Dynamic?Dynamic:N+1), Flags >	Diag (const Matrix< T, M, N, Flags > &A, Complex< T > z)
	Computes the (M+1)x(N+1) block diagonal matrix. More...

template<typename T , unsigned int Flags = 0>
Matrix< T, 2, 2, Flags >	Diag (Complex< T > z, Complex< T > w)
	Computes the 2x2 diagonal matrix. More...

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2, typename... Ts>
Matrix< T, Dynamic, Dynamic, Flags1 >	Diag (const Matrix< T, M1, N1, Flags1 > &a, const Matrix< T, M2, N2, Flags2 > &b, Ts &&... ts)

template<typename T , typename... Ts>
Matrix< T, Dynamic, Dynamic >	Diag (Complex< T > a, Complex< T > b, Ts &&... ts)

template<typename T , unsigned int Flags = 0>
SquareMatrix< T, Dynamic >	Diag (std::vector< Complex< T >> v)
	Dynamically computes the NxN diagonal matrix. More...

template<typename T , unsigned int Flags>
Matrix< T, Dynamic, Dynamic, Flags >	Diag (std::vector< Matrix< T, Dynamic, Dynamic, Flags >> As)
	Computes the PxQ diagonal matrix, where $P=\sum_{i=0}^{N-1}NumRows(As[i])$ and $Q=\sum_{i=0}^{N-1}NumColumns(As[i])$,. More...

template<typename T , size_t N, unsigned int Flags = 0>
SquareMatrix< T, N >	Diag (Vector< T, N > v)
	Computes the NxN diagonal matrix. More...

template<typename T , size_t N, unsigned int Flags = 0>
SquareMatrix< T, N >	Diag (RowVector< T, N > v)
	Computes the NxN diagonal matrix. More...

template<typename T , size_t N, unsigned int Flags = 0>
std::enable_if<(N >1\|\|N==Dynamic), SquareMatrix< T,(N==Dynamic?Dynamic:N-1), Flags > >::type	Companion (const Vector< T, N > &c)
	Computes the (N-1)x(N-1) companion matrix. More...

template<typename T , size_t N, unsigned int Flags = 0>
std::enable_if<(N >1\|\|N==Dynamic), SquareMatrix< T,(N==Dynamic?Dynamic:N-1), Flags > >::type	Companion (const RowVector< T, N > &c)
	Computes the (N-1)x(N-1) companion matrix. More...

template<typename T , size_t N, unsigned int Flags = 0>
SquareMatrix< T,(N==Dynamic?Dynamic:N+1), Flags >	Translation (const Vector< T, N > &t)
	Computes the (N+1)x(N+1) translation matrix T such that. More...

template<typename T , size_t N, unsigned int Flags = 0>
SquareMatrix< T, N, Flags >	Scaling (const Vector< T, N > &s)
	Computes the NxN scaling matrix S such that. More...

template<typename T , size_t N = 4, unsigned int Flags = 0>
std::enable_if<(N==3\|\|N==4), SquareMatrix< T, N, Flags > >::type	RotationX (T theta)
	Computes the 4x4 rotatation matrix. More...

template<typename T , size_t N = 4, unsigned int Flags = 0>
std::enable_if<(N==3\|\|N==4), SquareMatrix< T, N, Flags > >::type	RotationY (T theta)
	Computes the 4x4 rotatation matrix. More...

template<typename T , size_t N, unsigned int Flags = 0>
std::enable_if<(N==3\|\|N==4), SquareMatrix< T, N, Flags > >::type	RotationZ (T theta)
	Computes the 4x4 rotatation matrix. More...

template<size_t P, typename T , size_t N = 4, unsigned int Flags = 0>
std::enable_if<((N==3\|\|N==4)&&(P==3\|\|P==Dynamic)), SquareMatrix< T, N, Flags > >::type	Rotation (T theta, Vector< T, P > axis)
	Computes the 4x4 rotatation matrix about an axis. More...

template<size_t P, typename T , size_t N = 4, unsigned int Flags = 0>
std::enable_if<((N==3\|\|N==4)&&(P==4\|\|P==Dynamic)), SquareMatrix< T, N, Flags > >::type	Rotation (Vector< T, P > q)
	Computes the 4x4 rotatation matrix given a quaternion q. More...

template<typename T , unsigned int Flags = 0>
SquareMatrix< T, 4, Flags >	Orthographic (T left, T right, T bottom, T top, T near, T far)
	Computes the 4x4 orthographic projection matrix. More...

template<typename T , unsigned int Flags = 0>
SquareMatrix< T, 4, Flags >	Perspective (T fov, T aspect, T near, T far)
	Computes the 4x4 perspective projection matrix. More...

template<typename T , size_t N1, size_t N2, size_t N3, size_t N4, unsigned int Flags = 0>
std::enable_if<((N1==3\|\|N1==Dynamic)\|\|(N2==3\|\|N2==Dynamic)\|\|(N3==3\|\|N3==Dynamic)\|\|(N4==3\|\|N4==Dynamic)), SquareMatrix< T, 4, Flags > >::type	LookAt (Vector< T, N1 > right, Vector< T, N2 > up, Vector< T, N3 > dir, Vector< T, N4 > eye)
	Computes the 4x4 look at matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P>
std::enable_if<(P==N\|\|P==Dynamic\|\|N==Dynamic), Eigenpair< T, P > >::type	PowerIteration (const Matrix< T, M, N, Flags > &A, Vector< T, P > b0, unsigned int max_iterations)
	Performs power iteration on A. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P>
std::enable_if<(P==N\|\|P==Dynamic\|\|N==Dynamic), Vector< T, P > >::type	InverseIteration (const Matrix< T, M, N, Flags > &A, Vector< T, P > b0, Complex< T > mu, unsigned int max_iterations)
	Performs inverse iteration on A with guess mu. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Vector< T, N >	WielandtDeflationAlgorithm (const Matrix< T, M, N, Flags > &A, unsigned int max_iterations=100)
	Performs a combination of deflation and power iteration to get every eigenpair of A Since power iteration returns the largest eigenvalue, if we want to compute the eigenvalues we need to use deflation. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Vector< T, N >	Eigenvalues (const Matrix< T, M, N, Flags > &A)
	Calculates all eigenvalues of A. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
std::vector< Eigenpair< T, N > >	Eigen (const Matrix< T, M, N, Flags > &A)
	Calculates all eigenpairs of A. More...

template<typename T , size_t N>
void	VietaFormulaHelper (size_t data[], size_t start, size_t end, size_t index, size_t k, Complex< T > &sum, const Vector< T, N > &roots)

template<typename T , size_t M, size_t N, unsigned int Flags>
Vector< T,(N==Dynamic?Dynamic:N+1)>	CharPoly (const Matrix< T, M, N, Flags > &A)
	Computes the characteristic polynomial. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>
Matrix< T, M, N, Flags >	EntrywiseProduct (const Matrix< T, M, N, Flags > &A, const Matrix< T, P, Q, Flags2 > &B)
	Computes the matrix $C$ defined by $c_{ij}=a_{ij}b_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>
Matrix< T, M, N, Flags >	EntrywiseDivision (const Matrix< T, M, N, Flags > &A, const Matrix< T, P, Q, Flags2 > &B)
	Computes the matrix $C$ defined by $c_{ij}=a_{ij}/b_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
T	Norm (const Matrix< T, M, N, Flags > &A, size_t p=2)
	If $A$ is a vector, it computes the entrywise vector p-norm $\left(\sum_{i=0}^{N-1}\|a_i\|^p\right)^{1/p}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
T	EntrywiseNorm (const Matrix< T, M, N, Flags > &A, size_t p=2)
	Computes the entrywise p-norm $\left(\sum_{j=0}^{N-1}\sum_{i=0}^{M-1}\|a_{ij}\|^p\right)^{1/p}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
T	FrobeniusNorm (const Matrix< T, M, N, Flags > &A)
	Computes the Frobenius norm. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
T	MaxNorm (const Matrix< T, M, N, Flags > &A)
	Computes the max norm $\max_{ij}\|a_{ij}\|$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
T	InfinityNorm (const Matrix< T, M, N, Flags > &A)
	Computes the infinity norm $\max_{i=0,...M-1}\sum_{j=0}^{N-1}\|a_{ij}\|$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsReal (const Matrix< T, M, N, Flags > &A)
	Determines if a matrix is real or complex. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Abs (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix $B$ defined by $b_{ij}=\|a_{ij}\|$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Sign (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix $B$ defined by $b_{ij}=sgn(a_{ij})$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Arg (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix $B$ defined by $b_{ij}=arg(a_{ij})$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Conjugate (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix $B$ defined by $b_{ij}=\overline{a_{ij}}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Sqrt (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix $B$ defined by $b_{ij}=\sqrt{a_{ij}}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Exp (Matrix< T, M, N, Flags > A)
	If $A$ is not square, it computes the MxN Matrix $B$ defined by $b_{ij}=e^{a_{ij}}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Log (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix $B$ defined by $b_{ij}=\log a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Log (Matrix< T, M, N, Flags > A, T base)
	Computes the MxN Matrix $B$ defined by $b_{ij}=\log_{base} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Log (Matrix< T, M, N, Flags > A, Complex< T > base)
	Computes the MxN Matrix $B$ defined by $b_{ij}=\log_{base} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Pow (Matrix< T, M, N, Flags > A, Complex< T > power)
	If $A$ is not square, it computes the MxN Matrix $B$ defined by $b_{ij}=a_{ij}^{power}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags, typename U >
Matrix< T, M, N, Flags >	Pow (const Matrix< T, M, N, Flags > &A, U power)

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Mod (Matrix< T, M, N, Flags > A, Complex< T > z)
	Computes the MxN Matrix $B$ defined by $b_{ij}=a_{ij}\bmod z$. More...

template<typename T , size_t M, size_t N, unsigned int Flags, typename U >
Matrix< T, M, N, Flags >	Mod (Matrix< T, M, N, Flags > A, U y)
	Computes the MxN Matrix $B$ defined by $b_{ij}=a_{ij}\bmod y$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Sin (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\sin a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Cos (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\cos a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Tan (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\tan a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Csc (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\csc a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Sec (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\sec a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Cot (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\cot a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ASin (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\sin^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ACos (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\cos^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ATan (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\tan^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ACsc (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\csc^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ASec (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\sec^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ACot (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\cot^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Sinh (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\sinh a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Cosh (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\cosh a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Tanh (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\tanh a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Csch (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=csch a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Sech (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=sech a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Coth (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\coth a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ASinh (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\sinh^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ACosh (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\cosh^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ATanh (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\tanh^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ACsch (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=csch^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ASech (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=sech^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	ACoth (Matrix< T, M, N, Flags > A)
	Computes the MxN Matrix$B$ defined by $b_{ij}=\coth^{-1} a_{ij}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
std::vector< Vector< T, N > >	NullSpace (const Matrix< T, M, N, Flags > &A)
	Computes a basis for the null space of A. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
unsigned int	Nullity (const Matrix< T, M, N, Flags > &A)
	Computes the dimension of the A's null space. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
std::vector< Vector< T, N > >	ColumnSpace (const Matrix< T, M, N, Flags > &A)
	Computes a basis for the column space of A. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
unsigned int	Rank (const Matrix< T, M, N, Flags > &A)
	Computes the dimension of the A's column space. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P>
std::enable_if<(M==P\|\|P==Dynamic\|\|M==Dynamic), Vector< T, N > >::type	Solve (const Matrix< T, M, N, Flags > &A, const Vector< T, P > &b)
	Solves the matrix equation Ax=b for x. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P>
std::enable_if<(M==P\|\|P==Dynamic\|\|M==Dynamic), RowVector< T, M > >::type	Solve (const Matrix< T, M, N, Flags > &A, const RowVector< T, P > &b)
	Solves the matrix equation xA=b for x. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsVector (const Matrix< T, M, N, Flags > &A)
	Checks if a matrix is a row or column vector. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsSquare (Matrix< T, M, N, Flags > A)
	Checks if the matrix is squre. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsUpperTriangular (const Matrix< T, M, N, Flags > &A)
	Checks if A is upper triangular form. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsLowerTriangular (const Matrix< T, M, N, Flags > &A)
	Checks if A is lower triangular form. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsTriangular (const Matrix< T, M, N, Flags > &A)
	Checks if A is lower triangular form or upper triangular form. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsDiagonal (const Matrix< T, M, N, Flags > &A)
	Checks if A is a diagonal matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsIdentity (const Matrix< T, M, N, Flags > &A)
	Checks if A is the identity matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsCompanion (const Matrix< T, M, N, Flags > &A)
	Checks if A is a companion matrix. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsSymmetric (const Matrix< T, M, N, Flags > &A)
	Checks if a matrix is symmetric. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsHermitian (const Matrix< T, M, N, Flags > &A)
	Checks if a matrix is Hermitian. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsUpperHessenberg (const Matrix< T, M, N, Flags > &A)
	Checks if a matrix is upper Hessenberg. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsLowerHessenberg (const Matrix< T, M, N, Flags > &A)
	Checks if a matrix is lower Hessenberg. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsHessenberg (const Matrix< T, M, N, Flags > &a)
	Checks if a matrix is Hessenberg. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
bool	IsTridiagonal (const Matrix< T, M, N, Flags > &a)
	Checks if a matrix is Tridiagonal. More...

template<size_t P, typename T , size_t N>
Vector< T, P >	SubVector (const Vector< T, N > &v, size_t off=0)
	Constructs a subvector of a column vector. More...

template<typename T , size_t N>
Vector< T, Dynamic >	SubVector (const Vector< T, N > &v, size_t size, size_t off=0)
	Constructs a subvector of a column vector. More...

template<size_t P, typename T , size_t N>
RowVector< T, P >	SubVector (const RowVector< T, N > &v, size_t off=0)
	Constructs a subvector of a row vector. More...

template<typename T , size_t N>
RowVector< T, Dynamic >	SubVector (const RowVector< T, N > &v, size_t size, size_t off=0)
	Constructs a subvector of a row vector. More...

template<size_t P, typename T >
Vector< T, P >	SubVector (const Vector< T, 1 > &v, size_t off=0)
	Handles subvector in the size 1 case. More...

template<typename T >
Vector< T, Dynamic >	SubVector (const Vector< T, 1 > &v, size_t size, size_t off=0)
	Handles subvector in the size 1 case. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>
Complex< T >	Dot (const Matrix< T, M, N, Flags > &a, const Matrix< T, P, Q, Flags2 > &b)
	Computes the dot product of two vectors a and b both of size N, $\sum{i=0}^{N-1}\overline{a_i}b_i$. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	Normalize (Matrix< T, M, N, Flags > v)
	Normalize a vector, $v=\frac{v}{\\|v\\|}$. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>
Matrix< T, M, N, Flags >	Cross (const Matrix< T, M, N, Flags > &a, const Matrix< T, P, Q, Flags2 > &b)
	Computes the cross product of two vectors a and b both of length 3. More...

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>
Matrix< T, P, Q, Flags2 >	Proj (const Matrix< T, M, N, Flags > &v, const Matrix< T, P, Q, Flags2 > &onto)
	Computes the vector projection $\proj_{onto}v = \frac{Dot(onto,v)}{Dot(onto,onto)}onto$. More...

template<typename T , size_t N>
std::vector< Vector< T, N > >	GramSchmidt (const std::vector< Vector< T, N >> &v)
	Performs Gram-Schmidt on a list of column vectors. More...

template<typename T , size_t N>
std::vector< RowVector< T, N > >	GramSchmidt (const std::vector< RowVector< T, N >> &v)
	Performs Gram-Schmidt on a list of row vectors. More...

template<typename T , size_t M, size_t N, unsigned int Flags>
Matrix< T, M, N, Flags >	GramSchmidt (Matrix< T, M, N, Flags > A)
	Performs Gram-Schmidt on the columns of A. More...

template<typename T >
std::vector< Vector< T, 1 > >	GramSchmidt (const std::vector< Vector< T, 1 >> &v)
	Performs Gram-Schmidt on a list of length 1 vectors. More...

Variables
std::mt19937	random_number_generator

double	Tol = 0.00001

const unsigned int	Dynamic = 0

const unsigned int	RowMajor = 0x0000

const unsigned int	ColumnMajor = 0x0001

Typedef Documentation

◆ Complexd

using Linear::Complexd = typedef Complex<double>

◆ Complexf

using Linear::Complexf = typedef Complex<float>

◆ Complexi

using Linear::Complexi = typedef Complex<int>

◆ Matrix2d

using Linear::Matrix2d = typedef Matrix<double,2,2>

◆ Matrix2f

using Linear::Matrix2f = typedef Matrix<float,2,2>

◆ Matrix2ld

using Linear::Matrix2ld = typedef Matrix<long double,2,2>

◆ Matrix2x2d

using Linear::Matrix2x2d = typedef Matrix<double,2,2>

◆ Matrix2x2f

using Linear::Matrix2x2f = typedef Matrix<float,2,2>

◆ Matrix2x2ld

using Linear::Matrix2x2ld = typedef Matrix<long double,2,2>

◆ Matrix2x3d

using Linear::Matrix2x3d = typedef Matrix<double,2,3>

◆ Matrix2x3f

using Linear::Matrix2x3f = typedef Matrix<float,2,3>

◆ Matrix2x3ld

using Linear::Matrix2x3ld = typedef Matrix<long double,2,3>

◆ Matrix2x4d

using Linear::Matrix2x4d = typedef Matrix<double,2,4>

◆ Matrix2x4f

using Linear::Matrix2x4f = typedef Matrix<float,2,4>

◆ Matrix2x4ld

using Linear::Matrix2x4ld = typedef Matrix<long double,2,4>

◆ Matrix3d

using Linear::Matrix3d = typedef Matrix<double,3,3>

◆ Matrix3f

using Linear::Matrix3f = typedef Matrix<float,3,3>

◆ Matrix3ld

using Linear::Matrix3ld = typedef Matrix<long double,3,3>

◆ Matrix3x2d

using Linear::Matrix3x2d = typedef Matrix<double,3,2>

◆ Matrix3x2f

using Linear::Matrix3x2f = typedef Matrix<float,3,2>

◆ Matrix3x2ld

using Linear::Matrix3x2ld = typedef Matrix<long double,3,2>

◆ Matrix3x3d

using Linear::Matrix3x3d = typedef Matrix<double,3,3>

◆ Matrix3x3f

using Linear::Matrix3x3f = typedef Matrix<float,3,3>

◆ Matrix3x3ld

using Linear::Matrix3x3ld = typedef Matrix<long double,3,3>

◆ Matrix3x4d

using Linear::Matrix3x4d = typedef Matrix<double,3,4>

◆ Matrix3x4f

using Linear::Matrix3x4f = typedef Matrix<float,3,4>

◆ Matrix3x4ld

using Linear::Matrix3x4ld = typedef Matrix<long double,3,4>

◆ Matrix4d

using Linear::Matrix4d = typedef Matrix<double,4,4>

◆ Matrix4f

using Linear::Matrix4f = typedef Matrix<float,4,4>

◆ Matrix4ld

using Linear::Matrix4ld = typedef Matrix<long double,4,4>

◆ Matrix4x2d

using Linear::Matrix4x2d = typedef Matrix<double,4,2>

◆ Matrix4x2f

using Linear::Matrix4x2f = typedef Matrix<float,4,2>

◆ Matrix4x2ld

using Linear::Matrix4x2ld = typedef Matrix<long double,4,2>

◆ Matrix4x3d

using Linear::Matrix4x3d = typedef Matrix<double,4,3>

◆ Matrix4x3f

using Linear::Matrix4x3f = typedef Matrix<float,4,3>

◆ Matrix4x3ld

using Linear::Matrix4x3ld = typedef Matrix<long double,4,3>

◆ Matrix4x4d

using Linear::Matrix4x4d = typedef Matrix<double,4,4>

◆ Matrix4x4f

using Linear::Matrix4x4f = typedef Matrix<float,4,4>

◆ Matrix4x4ld

using Linear::Matrix4x4ld = typedef Matrix<long double,4,4>

◆ MatrixXd

using Linear::MatrixXd = typedef Matrix<double,Dynamic,Dynamic>

◆ MatrixXf

using Linear::MatrixXf = typedef Matrix<float,Dynamic,Dynamic>

◆ MatrixXld

using Linear::MatrixXld = typedef Matrix<long double,Dynamic,Dynamic>

◆ RowVector

template<typename T , size_t N>

using Linear::RowVector = typedef Matrix<T,1,N>

◆ RowVector2d

using Linear::RowVector2d = typedef RowVector<double,2>

◆ RowVector2f

using Linear::RowVector2f = typedef RowVector<float,2>

◆ RowVector2ld

using Linear::RowVector2ld = typedef RowVector<long double,2>

◆ RowVector3d

using Linear::RowVector3d = typedef RowVector<double,3>

◆ RowVector3f

using Linear::RowVector3f = typedef RowVector<float,3>

◆ RowVector3ld

using Linear::RowVector3ld = typedef RowVector<long double,3>

◆ RowVector4d

using Linear::RowVector4d = typedef RowVector<double,4>

◆ RowVector4f

using Linear::RowVector4f = typedef RowVector<float,4>

◆ RowVector4ld

using Linear::RowVector4ld = typedef RowVector<long double,4>

◆ RowVectorXd

using Linear::RowVectorXd = typedef RowVector<double,Dynamic>

◆ RowVectorXf

using Linear::RowVectorXf = typedef RowVector<float,Dynamic>

◆ RowVectorXld

using Linear::RowVectorXld = typedef RowVector<long double,Dynamic>

◆ SquareMatrix

template<typename T , size_t N, unsigned int Flags = 0>

using Linear::SquareMatrix = typedef Matrix<T,N,N,Flags>

◆ Vector

template<typename T , size_t N>

using Linear::Vector = typedef Matrix<T,N,1>

◆ Vector2d

using Linear::Vector2d = typedef Vector<double,2>

◆ Vector2f

using Linear::Vector2f = typedef Vector<float,2>

◆ Vector2ld

using Linear::Vector2ld = typedef Vector<long double,2>

◆ Vector3d

using Linear::Vector3d = typedef Vector<double,3>

◆ Vector3f

using Linear::Vector3f = typedef Vector<float,3>

◆ Vector3ld

using Linear::Vector3ld = typedef Vector<long double,3>

◆ Vector4d

using Linear::Vector4d = typedef Vector<double,4>

◆ Vector4f

using Linear::Vector4f = typedef Vector<float,4>

◆ Vector4ld

using Linear::Vector4ld = typedef Vector<long double,4>

◆ VectorXd

using Linear::VectorXd = typedef Vector<double,Dynamic>

◆ VectorXf

using Linear::VectorXf = typedef Vector<float,Dynamic>

◆ VectorXld

using Linear::VectorXld = typedef Vector<long double,Dynamic>

Enumeration Type Documentation

◆ SVDType

enum Linear::SVDType

Enumerator
FULL_SVD
THIN_SVD

Function Documentation

◆ Abs() [1/3]

template<typename T >

T Linear::Abs ( const Complex< T > & z )

Takes a complex number $z=a+bi$ and computes $|z|=\sqrt{a^2+b^2}$.

Parameters

z	Complex number.

Returns: Real number.

◆ Abs() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Abs ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix $B$ defined by $b_{ij}=|a_{ij}|$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Abs() [3/3]

template<typename T >

T Linear::Abs ( T x )

Takes a real number $x$ and computes $|x|$.

Parameters

x	Real number.

Returns: Real number.

◆ ACos() [1/3]

template<typename T >

Complex<T> Linear::ACos ( const Complex< T > & z )

Returns the complex number $\cos^{-1}z=\frac{1}{i}\log\left(z+i|1-z^2|^{1/2}e^{\frac{i}{2}arg(1-z^2)}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ACos() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ACos ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\cos^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ACos() [3/3]

template<typename T >

T Linear::ACos ( T x )

Returns the real number $\cos^{-1}x$.

Parameters

x	Real number.

Returns: Real number.

◆ ACosh() [1/3]

template<typename T >

Complex<T> Linear::ACosh ( const Complex< T > & z )

Returns the complex number $\cosh^{-1} z=\log\left(z+|z^2-1|^{1/2}e^{\frac{i}{2}arg(z^2-1)}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ACosh() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ACosh ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\cosh^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ACosh() [3/3]

template<typename T >

T Linear::ACosh ( T x )

Returns the real number $\cosh^{-1} x$.

Parameters

x	Real number.

Returns: Real number.

◆ ACot() [1/3]

template<typename T >

Complex<T> Linear::ACot ( const Complex< T > & z )

Returns the complex number $\cot^{-1}z=\frac{1}{2i}\log\left(\frac{z+i}{z-i}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ACot() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ACot ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\cot^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ACot() [3/3]

template<typename T >

T Linear::ACot ( T x )

Returns the real number $\cot^{-1}x=\tan^{-1}(1/x)$.

Parameters

x	Real number.

Returns: Real number.

◆ ACoth() [1/3]

template<typename T >

Complex<T> Linear::ACoth ( const Complex< T > & z )

Returns the complex number $\coth^{-1} z=\frac{1}{2}\log\left(\frac{z+1}{z-1}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ACoth() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ACoth ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\coth^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ACoth() [3/3]

template<typename T >

T Linear::ACoth ( T x )

Returns the real number $\coth^{-1} x=\tanh^{-1}(1/x)$.

Parameters

x	Real number.

Returns: Real number.

◆ ACsc() [1/3]

template<typename T >

Complex<T> Linear::ACsc ( const Complex< T > & z )

Returns the complex number $\csc^{-1}z=\sin^{-1}(1/z)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ACsc() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ACsc ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\csc^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ACsc() [3/3]

template<typename T >

T Linear::ACsc ( T x )

Returns the real number $\csc^{-1}x=\sin^{-1}(1/x)$.

Parameters

x	Real number.

Returns: Real number.

◆ ACsch() [1/3]

template<typename T >

Complex<T> Linear::ACsch ( const Complex< T > & z )

Returns the complex number $csch^{-1} z=\sinh^{-1}(1/z)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ACsch() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ACsch ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=csch^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ACsch() [3/3]

template<typename T >

T Linear::ACsch ( T x )

Returns the real number $csch^{-1} x=\sinh^{-1}(1/x)$.

Parameters

x	Real number.

Returns: Real number.

◆ Adjugate()

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Adjugate ( const Matrix< T, M, N, Flags > & A )

Computes the NxN adjugate B of a square matrix A defined by $b_{ij}=c_{ji}$ where $c_{ij}$ represents the (i,j)-cofactor of A.

If A is not square, an exception is thrown.

Parameters

A	MxN matrix

Returns: NxN matrix

◆ Arg() [1/2]

template<typename T >

T Linear::Arg ( const Complex< T > & z )

Takes a complex number $z=a+bi$ and computes $arg(z)=atan2(b,a)$.

Parameters

z	Complex number.

Returns: Real number.

◆ Arg() [2/2]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Arg ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix $B$ defined by $b_{ij}=arg(a_{ij})$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ASec() [1/3]

template<typename T >

Complex<T> Linear::ASec ( const Complex< T > & z )

Returns the complex number $\sec^{-1}z=\cos^{-1}(1/z)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ASec() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ASec ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\sec^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ASec() [3/3]

template<typename T >

T Linear::ASec ( T x )

Returns the real number $\sec^{-1}x=\cos^{-1}(1/x)$.

Parameters

x	Real number.

Returns: Real number.

◆ ASech() [1/3]

template<typename T >

Complex<T> Linear::ASech ( const Complex< T > & z )

Returns the complex number $sech^{-1} z=\cosh^{-1}(1/z)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ASech() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ASech ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=sech^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ASech() [3/3]

template<typename T >

T Linear::ASech ( T x )

Returns the real number $sech^{-1} x=\cosh^{-1}(1/x)$.

Parameters

x	Real number.

Returns: Real number.

◆ ASin() [1/3]

template<typename T >

Complex<T> Linear::ASin ( const Complex< T > & z )

Returns the complex number $\sin^{-1}z=\frac{1}{i}\log\left(iz+|1-z^2|^{1/2}e^{\frac{i}{2}arg(1-z^2)}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ASin() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ASin ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\sin^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ASin() [3/3]

template<typename T >

T Linear::ASin ( T x )

Returns the real number $\sin^{-1}x$.

Parameters

x	Real number.

Returns: Real number.

◆ ASinh() [1/3]

template<typename T >

Complex<T> Linear::ASinh ( const Complex< T > & z )

Returns the complex number $\sinh^{-1} z=\log\left(z+|1+z^2|^{1/2}e^{\frac{i}{2}arg(1+z^2)}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ASinh() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ASinh ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\sinh^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ASinh() [3/3]

template<typename T >

T Linear::ASinh ( T x )

Returns the real number $\sinh^{-1} x$.

Parameters

x	Real number.

Returns: Real number.

◆ ATan() [1/3]

template<typename T >

Complex<T> Linear::ATan ( const Complex< T > & z )

Returns the complex number $\tan^{-1}z=\frac{1}{2i}\log\left(\frac{i-z}{i+z}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ATan() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ATan ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\tan^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ATan() [3/3]

template<typename T >

T Linear::ATan ( T x )

Returns the real number $\tan^{-1}x$.

Parameters

x	Real number.

Returns: Real number.

◆ ATanh() [1/3]

template<typename T >

Complex<T> Linear::ATanh ( const Complex< T > & z )

Returns the complex number $\tanh^{-1} z=\frac{1}{2}\log\left(\frac{1+z}{1-z}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ ATanh() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::ATanh ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\tanh^{-1} a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ATanh() [3/3]

template<typename T >

T Linear::ATanh ( T x )

Returns the real number $\tanh^{-1} x$.

Parameters

x	Real number.

Returns: Real number.

◆ Augmented()

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2>

std::enable_if<(M1==M2\|\|M1==Dynamic\|\|M2==Dynamic), Matrix<T,M1,(N1==Dynamic\|\|N2==Dynamic?Dynamic:N1+N2),Flags1> >::type Linear::Augmented	(	const Matrix< T, M1, N1, Flags1 > &	left,
		const Matrix< T, M2, N2, Flags2 > &	right
	)

Creates the M1x(N1+N2) augmented matrix.

\[ \begin{bmatrix} left & \mid & right \end{bmatrix}. \]

If the two matrices have differing numbers of rows, an exception is thrown.

Parameters

left	M1xN1 Matrix
right	M2xN2 Matrix

Returns: M1x(N1+N2) Matrix

◆ Basis() [1/2]

template<typename T , size_t M, size_t N, unsigned int Flags = 0>

std::enable_if<(M>0&&N>0), Matrix<T,M,N,Flags> >::type Linear::Basis	(	size_t	i,
		size_t	j
	)

Creates an MxN basis matrix $A$ where all entries are set to 0 except $a_{ij}=1$.

If i or j are out of bounds, an exception is thrown.

Example: A is the 2x2 matrix {{0,1}, {0,0}}

Matrix2d A = Basis<double, 2, 2>(0, 1);

Parameters

T	Type
M	Number of rows
N	Number of columns
Flags	Flags to pass to the matrix (default = row major)
i	Row index
j	Column index

Returns: MxN Matrix

◆ Basis() [2/2]

template<typename T , unsigned int Flags = 0>

Matrix<T,Dynamic,Dynamic,Flags> Linear::Basis	(	size_t	nrows,
		size_t	ncols,
		size_t	i,
		size_t	j
	)

Dynamically creates an nrowsxncols basis matrix $A$ where all entries are set to 0 except $a_{ij}=1$.

If i or j are out of bounds, an exception is thrown.

Example: A is the 2x2 matrix {{0,1}, {0,0}}

MatrixXd A = Basis<double>(2, 2, 0, 1);

Parameters

T	Type
Flags	Flags to pass to the matrix (default = row major)
nrows	Number of rows
ncols	Number of columns
i	Row index
j	Column index

Returns: nrowsxncols Matrix

◆ Block() [1/3]

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q>

std::enable_if<((M==P\|\|M==Dynamic\|\|P==Dynamic)&&(N==Q\|\|N==Dynamic\|\|Q==Dynamic)), Matrix<T,(M==Dynamic?Dynamic:M+1),(N==Dynamic?Dynamic:N+1),Flags> >::type Linear::Block	(	Complex< T >	tl,
		const RowVector< T, Q > &	tr,
		const Vector< T, P > &	bl,
		const Matrix< T, M, N, Flags > &	br
	)

Creates the (M+1)x(N+1) block matrix.

\[ \begin{bmatrix} tl & tr \\ bl & br \end{bmatrix}. \]

If $P\ne M$ or $Q\ne N$, then an exception is thrown.

Parameters

tl	Complex number
tr	Row vector of length Q
bl	Vector of length P
br	MxN Matrix

Returns: (M+1)x(N+1) Matrix

◆ Block() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q>

std::enable_if<((M==P\|\|M==Dynamic\|\|P==Dynamic)&&(N==Q\|\|N==Dynamic\|\|Q==Dynamic)), Matrix<T,(M==Dynamic?Dynamic:M+1),(N==Dynamic?Dynamic:N+1),Flags> >::type Linear::Block	(	const Matrix< T, M, N, Flags > &	tl,
		const Vector< T, P > &	tr,
		const RowVector< T, Q > &	bl,
		Complex< T >	br
	)

Creates the (M+1)x(N+1) block matrix.

\[ \begin{bmatrix} tl & tr \\ bl & br \end{bmatrix}. \]

If $P\ne M$ or $Q\ne N$, then an exception is thrown.

Parameters

tl	MxN Matrix
tr	Vector of length P
bl	Row vector of length Q
br	Complex number

Returns: (M+1)x(N+1) Matrix

◆ Block() [3/3]

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2, size_t M3, size_t N3, unsigned int Flags3, size_t M4, size_t N4, unsigned int Flags4>

std::enable_if<((M1+M3==M2+M4\|\|M1==Dynamic\|\|M2==Dynamic\|\|M3==Dynamic\|\|M4==Dynamic)&&(N1+N2==N3+N4\|\|N1==Dynamic\|\|N2==Dynamic\|\|N3==Dynamic\|\|N4==Dynamic)), Matrix<T,(M1==Dynamic\|\|M2==Dynamic\|\|M3==Dynamic\|\|M4==Dynamic?Dynamic:M1+M3),(N1==Dynamic\|\|N2==Dynamic\|\|N3==Dynamic\|\|N4==Dynamic?Dynamic:N1+N2),Flags1> >::type Linear::Block	(	const Matrix< T, M1, N1, Flags1 > &	tl,
		const Matrix< T, M2, N2, Flags2 > &	tr,
		const Matrix< T, M3, N3, Flags3 > &	bl,
		const Matrix< T, M4, N4, Flags4 > &	br
	)

Creates the (M1+M3)x(N1+N2) block matrix.

\[ \begin{bmatrix} tl & tr \\ bl & br \end{bmatrix}. \]

If $M1+M3\ne M2+M4$ or $N1+N2\ne N3+N4$, then an exception is thrown.

Parameters

tl	M1xN1 Matrix
tr	M2xN2 Matrix
bl	M3xN3 Matrix
br	M4xN4 Matrix

Returns: (M1+M3)x(N1+N2) Matrix

◆ Ceil() [1/2]

template<typename T >

Complex<T> Linear::Ceil ( const Complex< T > & z )

Returns the complex number $\lceil a\rceil+\lceil b\rceil i$ where $z=a+bi$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Ceil() [2/2]

template<typename T >

T Linear::Ceil ( T x )

Returns the real number $\lceil x\rceil$.

Parameters

x	Real number.

Returns: Real number.

◆ CharPoly()

template<typename T , size_t M, size_t N, unsigned int Flags>

Vector<T,(N==Dynamic?Dynamic:N+1)> Linear::CharPoly ( const Matrix< T, M, N, Flags > & A )

Computes the characteristic polynomial.

This returns the coefficients as a vector c of length N+1 such that $\det(A-tI)=c_0+c_1t+\dots+c_{N-1}t^{N-1}+c_Nt^N$.

Parameters

A	MxN matrix

Returns: Coefficients of characteristic polynomial

◆ Cofactor()

template<typename T , size_t M, size_t N, unsigned int Flags>

Complex<T> Linear::Cofactor	(	const Matrix< T, M, N, Flags > &	A,
		size_t	i,
		size_t	j
	)

Computes the (i,j)-cofactor of a square matrix defined by $(-1)^{i+j}\det(A_{ij})$ where $A_{ij}$ is the (N-1)x(N-1) matrix obtained by deleting the ith row and jth column.

If A is not square, an exception is thrown. Likewise if i or j is out-of-bounds an exception is thrown.

Parameters

A	MxN matrix
i	Row index
j	Column index

Returns: Complex number

◆ ColumnSpace()

template<typename T , size_t M, size_t N, unsigned int Flags>

std::vector<Vector<T,N> > Linear::ColumnSpace ( const Matrix< T, M, N, Flags > & A )

Computes a basis for the column space of A.

The column space of a matrix is the set of vectors v such that Ax=v for some vector x.

Parameters

A	MxN matrix

Returns: List of vectors v such that $colsp(A)=span\{v[0],\dots,v[len(v)-1]\}$

◆ Companion() [1/2]

template<typename T , size_t N, unsigned int Flags = 0>

std::enable_if<(N>1||N==Dynamic), SquareMatrix<T,(N==Dynamic?Dynamic:N-1),Flags> >::type Linear::Companion ( const RowVector< T, N > & c )

Computes the (N-1)x(N-1) companion matrix.

\[ C = \begin{bmatrix} 0 & 0 & \dots & 0 & -c_0/c_{N-1} \\ 1 & 0 & \dots & 0 & -c_1/c_{N-1} \\ 0 & 1 & \dots & 0 & -c_2/c_{N-1} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_{N-2}/c_{N-1} \end{bmatrix} \]

such that $\det(C-tI)=c_0/c_{N-1}+c_1/c_{N-1}t+\dots+c_{N-2}/c_{N-1}t^{N-2}+t^{N-1}$.

Parameters

Flags	Flags to pass to the matrix (default = row major)
c	Row vector of size N

Returns: (N-1)x(N-1) Matrix

◆ Companion() [2/2]

template<typename T , size_t N, unsigned int Flags = 0>

std::enable_if<(N>1||N==Dynamic), SquareMatrix<T,(N==Dynamic?Dynamic:N-1),Flags> >::type Linear::Companion ( const Vector< T, N > & c )

Computes the (N-1)x(N-1) companion matrix.

\[ C = \begin{bmatrix} 0 & 0 & \dots & 0 & -c_0/c_{N-1} \\ 1 & 0 & \dots & 0 & -c_1/c_{N-1} \\ 0 & 1 & \dots & 0 & -c_2/c_{N-1} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & -c_{N-2}/c_{N-1} \end{bmatrix} \]

such that $\det(C-tI)=c_0/c_{N-1}+c_1/c_{N-1}t+\dots+c_{N-2}/c_{N-1}t^{N-2}+t^{N-1}$.

Parameters

Flags	Flags to pass to the matrix (default = row major)
c	Vector of size N

Returns: (N-1)x(N-1) Matrix

◆ Conjugate() [1/2]

template<typename T >

Complex<T> Linear::Conjugate ( const Complex< T > & z )

Takes a complex number $z=a+bi$ and computes it's conjugate $\bar{z}=a-bi$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Conjugate() [2/2]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Conjugate ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix $B$ defined by $b_{ij}=\overline{a_{ij}}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ ConjugateTranspose()

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,N,M,Flags> Linear::ConjugateTranspose ( Matrix< T, M, N, Flags > A )

Returns the NxM matrix $B=A^*$ defined by $b_{ij}=\overline{a_{ji}}$.

Parameters

A	MxN matrix

Returns: NxM matrix

◆ Constant() [1/2]

template<typename T , size_t M, size_t N, unsigned int Flags = 0>

Matrix<T,M,N,Flags> Linear::Constant ( Complex< T > z )

Creates an MxN matrix with every entry set to z.

Example: A is the 2x3 matrix {{1+1i,1+1i,1+1i}, {1+1i,1+1i,1+1i}}

Matrix<double,2,3> A = Constant<double, 2, 3>(Complexd(1,1));

Parameters

T	Type
M	Number of rows
N	Number of columns
Flags	Flags to pass to the matrix (default = row major)
z	Complex number

Returns: MxN Matrix

◆ Constant() [2/2]

template<typename T , unsigned int Flags = 0>

Matrix<T,Dynamic,Dynamic,Flags> Linear::Constant	(	size_t	nrows,
		size_t	ncols,
		Complex< T >	z
	)

Dynamically creates an nrowsxncols matrix with every entry set to v.

Example: A is the 2x3 matrix {{1+1i,1+1i,1+1i}, {1+1i,1+1i,1+1i}}

MatrixXd A = Constant<double>(2, 3, Complexd(1,1));

Parameters

T	Type
Flags	Flags to pass to the matrix (default = row major)
nrows	Number of rows
ncols	Number of columns
z	Complex number

Returns: nrowsxncols Matrix

◆ Cos() [1/3]

template<typename T >

Complex<T> Linear::Cos ( const Complex< T > & z )

Returns the complex number $\cos z=\frac{e^{iz}+e^{-iz}}{2}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Cos() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Cos ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\cos a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Cos() [3/3]

template<typename T >

T Linear::Cos ( T x )

Returns the real number $\cos x$.

Parameters

x	Real number.

Returns: Real number.

◆ Cosh() [1/3]

template<typename T >

Complex<T> Linear::Cosh ( const Complex< T > & z )

Returns the complex number $\cosh z=\frac{e^z+e^{-z}}{2}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Cosh() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Cosh ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\cosh a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Cosh() [3/3]

template<typename T >

T Linear::Cosh ( T x )

Returns the real number $\cosh x$.

Parameters

x	Real number.

Returns: Real number.

◆ Cot() [1/3]

template<typename T >

Complex<T> Linear::Cot ( const Complex< T > & z )

Returns the complex number $\cot z=i\left(\frac{e^{2iz}+1}{e^{2iz}-1}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Cot() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Cot ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\cot a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Cot() [3/3]

template<typename T >

T Linear::Cot ( T x )

Returns the real number $\cot x=1/\tan x$.

Parameters

x	Real number.

Returns: Real number.

◆ Coth() [1/3]

template<typename T >

Complex<T> Linear::Coth ( const Complex< T > & z )

Returns the complex number $\coth z=\frac{e^z+e^{-z}}{e^z-e^{-z}}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Coth() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Coth ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\coth a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Coth() [3/3]

template<typename T >

T Linear::Coth ( T x )

Returns the real number $\coth x = 1/\tanh x$.

Parameters

x	Real number.

Returns: Real number.

◆ Cross()

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>

Matrix<T,M,N,Flags> Linear::Cross	(	const Matrix< T, M, N, Flags > &	a,
		const Matrix< T, P, Q, Flags2 > &	b
	)

Computes the cross product of two vectors a and b both of length 3.

The cross product is defined by

\[a\times b = (a_0i+a_1j+a_2k)\times (b_0i+b_1j+b_2k)=(a_1b_2-a_2b_1)i + (a_2b_0-a_0b_2)j + (a_0b_1-a_1b_0)k.\]

If either a or b is not a vector, or they aren't length 3, an exception is thrown.

Example: c is the column vector {-3,6,-3}

Vector3d a = {1, 2, 3};
Vector3d b = {4, 5, 6};
Vector3d c = Cross(a,b);

Parameters

a	Row/column vector
b	Row/column vector

Returns: Column vector if a is a column vector, otherwise row vector.

◆ Csc() [1/3]

template<typename T >

Complex<T> Linear::Csc ( const Complex< T > & z )

Returns the complex number $\csc z=\frac{2i}{e^{iz}-e^{-iz}}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Csc() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Csc ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\csc a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Csc() [3/3]

template<typename T >

T Linear::Csc ( T x )

Returns the real number $\csc x=1/\sin x$.

Parameters

x	Real number.

Returns: Real number.

◆ Csch() [1/3]

template<typename T >

Complex<T> Linear::Csch ( const Complex< T > & z )

Returns the complex number $csch z=\frac{2}{e^z-e^{-z}}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Csch() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Csch ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=csch a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Csch() [3/3]

template<typename T >

T Linear::Csch ( T x )

Returns the real number $csch x = 1/\sinh x$.

Parameters

x	Real number.

Returns: Real number.

◆ Determinant()

template<typename T , size_t M, size_t N, unsigned int Flags>

Complex<T> Linear::Determinant ( const Matrix< T, M, N, Flags > & A )

Computes the determinant of a square matrix defined by $\sum_{i=0}^{N-1}(-1)^ia_{0i}\det(A_{0i})$ where $A_{0i}$ is the (N-1)x(N-1) matrix obtained by deleting the 0th row and ith column.

If A is not square, an exception is thrown.

Parameters

A	MxN matrix

Returns: Complex number

◆ Diag() [1/9]

template<typename T , typename... Ts>

Matrix<T,Dynamic,Dynamic> Linear::Diag	(	Complex< T >	a,
		Complex< T >	b,
		Ts &&...	ts
	)

◆ Diag() [2/9]

template<typename T , unsigned int Flags = 0>

Matrix<T,2,2,Flags> Linear::Diag	(	Complex< T >	z,
		Complex< T >	w
	)

Computes the 2x2 diagonal matrix.

\[ \begin{bmatrix} z & 0 \\ 0 & w \end{bmatrix}. \]

Parameters

Flags	Flags to pass to the matrix (default = row major)
z	Complex number
w	Complex number

Returns: 2x2 Matrix

◆ Diag() [3/9]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,(M==Dynamic?Dynamic:M+1),(N==Dynamic?Dynamic:N+1),Flags> Linear::Diag	(	const Matrix< T, M, N, Flags > &	A,
		Complex< T >	z
	)

Computes the (M+1)x(N+1) block diagonal matrix.

\[ \begin{bmatrix} A & 0 \\ 0 & z \end{bmatrix}. \]

Parameters

A	MxN Matrix
z	Complex number

Returns: (M+1)x(N+1) Matrix

◆ Diag() [4/9]

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2>

Matrix<T,(M1==Dynamic\|\|M2==Dynamic?Dynamic:M1+M2),(N1==Dynamic\|\|N2==Dynamic?Dynamic:N1+N2),Flags1> Linear::Diag	(	const Matrix< T, M1, N1, Flags1 > &	A,
		const Matrix< T, M2, N2, Flags2 > &	B
	)

Computes the (M1+M2)x(N1+N2) block diagonal matrix.

\[ \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix}. \]

Parameters

A	M1xN1 Matrix
B	M2xN2 Matrix

Returns: (M1+M2)x(N1+N2) Matrix

◆ Diag() [5/9]

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2, typename... Ts>

Matrix<T,Dynamic,Dynamic,Flags1> Linear::Diag	(	const Matrix< T, M1, N1, Flags1 > &	a,
		const Matrix< T, M2, N2, Flags2 > &	b,
		Ts &&...	ts
	)

◆ Diag() [6/9]

template<typename T , size_t N, unsigned int Flags = 0>

SquareMatrix<T,N> Linear::Diag ( RowVector< T, N > v )

Computes the NxN diagonal matrix.

\[ diag(v_0,\dots,v_{N-1})=\begin{bmatrix} v_0 & & \\ & \ddots & \\ & & v_{N-1} \end{bmatrix}. \]

Parameters

Flags	Flags to pass to the matrix (default = row major)
v	Row vector of size N

Returns: NxN Matrix

◆ Diag() [7/9]

template<typename T , unsigned int Flags = 0>

SquareMatrix<T,Dynamic> Linear::Diag ( std::vector< Complex< T >> v )

Dynamically computes the NxN diagonal matrix.

\[ diag(v[0],\dots,v[N-1])=\begin{bmatrix} v[0] & & \\ & \ddots & \\ & & v[N-1] \end{bmatrix}. \]

Parameters

Flags	Flags to pass to the matrix (default = row major)
v	List of N Complex numbers

Returns: NxN Matrix

◆ Diag() [8/9]

template<typename T , unsigned int Flags>

Matrix<T,Dynamic,Dynamic,Flags> Linear::Diag ( std::vector< Matrix< T, Dynamic, Dynamic, Flags >> As )

Computes the PxQ diagonal matrix, where $P=\sum_{i=0}^{N-1}NumRows(As[i])$ and $Q=\sum_{i=0}^{N-1}NumColumns(As[i])$,.

\[ diag(As[0],\dots,As[N-1])=\begin{bmatrix} As[0] & & \\ & \ddots & \\ & & As[N-1] \end{bmatrix}. \]

Parameters

Flags	Flags to pass to the matrix (default = row major)
As	List of N matrices

Returns: PxQ Matrix

◆ Diag() [9/9]

template<typename T , size_t N, unsigned int Flags = 0>

SquareMatrix<T,N> Linear::Diag ( Vector< T, N > v )

Computes the NxN diagonal matrix.

\[ diag(v_0,\dots,v_{N-1}) = \begin{bmatrix} v_0 & & \\ & \ddots & \\ & & v_{N-1} \end{bmatrix}. \]

Parameters

Flags	Flags to pass to the matrix (default = row major)
v	Vector of size N

Returns: NxN Matrix

◆ Dot()

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>

Complex<T> Linear::Dot	(	const Matrix< T, M, N, Flags > &	a,
		const Matrix< T, P, Q, Flags2 > &	b
	)

Computes the dot product of two vectors a and b both of size N, $\sum{i=0}^{N-1}\overline{a_i}b_i$.

If either a or b is not a vector, or they have differing sizes, an exception is thrown.

Example: z=1*3+2*4=11

Vector2d a = {1, 2};
Vector2d b = {3, 4};
Complexd z = Dot(a,b);

Parameters

a	Row/column vector
b	Row/column vector

Returns: Complex number

◆ Eigen()

template<typename T , size_t M, size_t N, unsigned int Flags>

std::vector<Eigenpair<T,N> > Linear::Eigen ( const Matrix< T, M, N, Flags > & A )

Calculates all eigenpairs of A.

This function handles various cases. First, if the matrix is either upper or lower triangular, then the eigenvalues are simply across the diagonal. Allowing us to pull them directly out of the matrix and use either the Nullspace or InverseIteration to calculate their corresponding eigenvectors. Second, if the matrix is 2x2, then a easy direct formula exists to calculate the eigenvalues. Namely $(tr(A)\pm\sqrt{tr(A)^2-4det(A)})/2$. Third, if neither of the first two cases handles the matrix, we attempt to calculate the Schur decomposition of A. If that fails, we resort to calling WielandtDeflationAlgorithm.

Parameters

A	MxN matrix

Returns: List of Eigenpairs

◆ Eigenvalues()

template<typename T , size_t M, size_t N, unsigned int Flags>

Vector<T,N> Linear::Eigenvalues ( const Matrix< T, M, N, Flags > & A )

Calculates all eigenvalues of A.

This function handles various cases. First, if the matrix is either upper or lower triangular, then the eigenvalues are simply across the diagonal. Allowing us to pull them directly out of the matrix and use either the Nullspace or InverseIteration to calculate their corresponding eigenvectors. Second, if the matrix is 2x2, then a easy direct formula exists to calculate the eigenvalues. Namely $(tr(A)\pm\sqrt{tr(A)^2-4det(A)})/2$. Third, if neither of the first two cases handles the matrix, we attempt to calculate the Schur decomposition of A. If that fails, we resort to calling WielandtDeflationAlgorithm.

Parameters

A	MxN matrix

Returns: Column vector of eigenvalues.

◆ EntrywiseDivision()

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>

Matrix<T,M,N,Flags> Linear::EntrywiseDivision	(	const Matrix< T, M, N, Flags > &	A,
		const Matrix< T, P, Q, Flags2 > &	B
	)

Computes the matrix $C$ defined by $c_{ij}=a_{ij}/b_{ij}$.

If $M\ne P$ or $N\ne Q$ an exception is thrown.

Parameters

A	MxN Matrix
B	PxQ Matrix

Returns: MxN Matrix

◆ EntrywiseNorm()

template<typename T , size_t M, size_t N, unsigned int Flags>

T Linear::EntrywiseNorm	(	const Matrix< T, M, N, Flags > &	A,
		size_t	p = `2`
	)

Computes the entrywise p-norm $\left(\sum_{j=0}^{N-1}\sum_{i=0}^{M-1}|a_{ij}|^p\right)^{1/p}$.

Parameters

A	MxN Matrix
p	Real number (default = 2)

Returns: Real number

◆ EntrywiseProduct()

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>

Matrix<T,M,N,Flags> Linear::EntrywiseProduct	(	const Matrix< T, M, N, Flags > &	A,
		const Matrix< T, P, Q, Flags2 > &	B
	)

Computes the matrix $C$ defined by $c_{ij}=a_{ij}b_{ij}$.

If $M\ne P$ or $N\ne Q$ an exception is thrown.

Parameters

A	MxN Matrix
B	PxQ Matrix

Returns: MxN Matrix

◆ Exp() [1/3]

template<typename T >

Complex<T> Linear::Exp ( const Complex< T > & z )

Takes a complex number $z$ and computes $e^z$.

This is done using Euler's formula. Thus $e^z=e^{a+bi}=e^ae^{bi}=e^a(\cos b+i\sin b)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Exp() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Exp ( Matrix< T, M, N, Flags > A )

If $A$ is not square, it computes the MxN Matrix $B$ defined by $b_{ij}=e^{a_{ij}}$.

Otherwise it computes the matrix exponential $e^A=\sum_{k=0}^\infty\frac{1}{k!}A^k$. If $A=VDV^{-1}$ with $D=diag(d_1,\dots,d_N)$ a diagonal matrix, then $e^{A}=Ve^{D}V^{-1}$ and $e^D=diag(e^{d_1},\dots,e^{d_N})$. If no such decomposition can be found, it simply estimates the series upto the first 10 terms.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Exp() [3/3]

template<typename T >

T Linear::Exp ( T x )

Takes a real number $x$ and computes $e^x$.

Parameters

x	Real number.

Returns: Real number.

◆ Floor() [1/2]

template<typename T >

Complex<T> Linear::Floor ( const Complex< T > & z )

Returns the complex number $\lfloor a\rfloor+\lfloor b\rfloor i$ where $z=a+bi$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Floor() [2/2]

template<typename T >

T Linear::Floor ( T x )

Returns the real number $\lfloor x\rfloor$.

Parameters

x	Real number.

Returns: Real number.

◆ FrobeniusNorm()

template<typename T , size_t M, size_t N, unsigned int Flags>

T Linear::FrobeniusNorm ( const Matrix< T, M, N, Flags > & A )

Computes the Frobenius norm.

This is identical to EntrywiseNorm(A,2).

Parameters

A	MxN Matrix

Returns: Real number

◆ GramSchmidt() [1/4]

template<typename T , size_t N>

std::vector<RowVector<T,N> > Linear::GramSchmidt ( const std::vector< RowVector< T, N >> & v )

Performs Gram-Schmidt on a list of row vectors.

If all vectors don't have the same length, an exception is thrown.

Example: vectors is the list of row vectors {0.477,0.894} and {0.894,-0.477}

RowVector2d v1 = { 1, 2 };
RowVector2d v2 = { 3, 4 };
std::vector<RowVector2d> vectors = { v1, v2 };
vectors = GramSchmidt(vectors);

Parameters

v	List of row vectors

Returns: List of row vectors

◆ GramSchmidt() [2/4]

template<typename T >

std::vector<Vector<T,1> > Linear::GramSchmidt ( const std::vector< Vector< T, 1 >> & v )

Performs Gram-Schmidt on a list of length 1 vectors.

Example: vectors is the list of length 1 vectors {1} and {1}

Vector<double, 1> v1 = { 1 };
Vector<double, 1> v2 = { 2 };
std::vector<Vector<double,1>> vectors = { v1, v2 };
vectors = GramSchmidt(vectors);

Parameters

v	List of vectors all of length 1

Returns: List of vectors all of length 1

◆ GramSchmidt() [3/4]

template<typename T , size_t N>

std::vector<Vector<T,N> > Linear::GramSchmidt ( const std::vector< Vector< T, N >> & v )

Performs Gram-Schmidt on a list of column vectors.

If all vectors don't have the same length, an exception is thrown.

Example: vectors is the list of column vectors {0.477,0.894} and {0.894,-0.477}

Vector2d v1 = { 1, 2 };
Vector2d v2 = { 3, 4 };
std::vector<Vector2d> vectors = { v1, v2 };
vectors = GramSchmidt(vectors);

Parameters

v	List of column vectors

Returns: List of column vectors

◆ GramSchmidt() [4/4]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::GramSchmidt ( Matrix< T, M, N, Flags > A )

Performs Gram-Schmidt on the columns of A.

Example: B is the 2x2 matrix {{0.477,0.894}, {0.894,-0.477}}

Matrix2d A = { {1, 3}, {2, 4} };
Matrix2d B = GramSchmidt(A);

Parameters

A	MxN Matrix

Returns: MxN matrix

◆ Householder() [1/2]

template<typename T , size_t N, unsigned int Flags = 0>

SquareMatrix<T,N,Flags> Linear::Householder	(	const Vector< T, N > &	x,
		size_t	k
	)

Creates an NxN Householder transformation $H$ such that $Hx$ eliminates the last k elements.

Parameters

Flags	Flags to pass to the matrix (default = row major)
x	Column vector of length N
k	Number of elements to zero out

Returns: NxN Matrix

◆ Householder() [2/2]

template<typename T , size_t N, unsigned int Flags = 0>

SquareMatrix<T,N,Flags> Linear::Householder ( Vector< T, N > v )

Creates an NxN Householder transformation $H$ defined by $H = I - 2vv^*.$.

Parameters

Flags	Flags to pass to the matrix (default = row major)
v	Column vector of length N

Returns: NxN Matrix

◆ Identity() [1/2]

template<typename T , size_t N, unsigned int Flags = 0>

SquareMatrix<T,N,Flags> Linear::Identity ( )

Creates the NxN Identity matrix.

Example: I is the 3x3 matrix {{1,0,0}, {0,1,0}, {0,0,1}}

Matrix3d I = Identity<double, 3>();

Parameters

T	Type
N	Size of matrix.
Flags	Flags to pass to the matrix (default = row major)

Returns: NxN Matrix

◆ Identity() [2/2]

template<typename T , unsigned int Flags = 0>

SquareMatrix<T,Dynamic,Flags> Linear::Identity ( size_t n )

Dynamically creates the nxn Identity matrix.

Example: I is the 3x3 matrix {{1,0,0}, {0,1,0}, {0,0,1}}

MatrixXd I = Identity<double>(3);

Parameters

T	Type
Flags	Flags to pass to the matrix (default = row major)
n	Size of matrix

Returns: nxn Matrix

◆ InfinityNorm()

template<typename T , size_t M, size_t N, unsigned int Flags>

T Linear::InfinityNorm ( const Matrix< T, M, N, Flags > & A )

Computes the infinity norm $\max_{i=0,...M-1}\sum_{j=0}^{N-1}|a_{ij}|$.

Parameters

A	MxN Matrix

Returns: Real number

◆ Inverse()

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Inverse ( const Matrix< T, M, N, Flags > & A )

Computes the NxN inverse $A^{-1}=\frac{1}{\det A}Adj$ of a square matrix A where Adj is the adjugate of A.

If A is not square or if A is singular, an exception is thrown.

Parameters

A	MxN matrix

Returns: NxN matrix

◆ InverseIteration()

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P>

std::enable_if<(P==N\|\|P==Dynamic\|\|N==Dynamic), Vector<T,P> >::type Linear::InverseIteration	(	const Matrix< T, M, N, Flags > &	A,
		Vector< T, P >	b0,
		Complex< T >	mu,
		unsigned int	max_iterations
	)

Performs inverse iteration on A with guess mu.

Inverse iteration, defined by the sequence $b_{k+1}=\frac{(A-\mu I)^{-1}b_k}{\|(A-\mu I)^{-1}b_k\|}$ is a simple algorithm that computes the eigenvalue of A closest to mu along with it's corresponding eigenvector. Convergence rate depends on how close mu is.

More information: https://en.wikipedia.org/wiki/Inverse_iteration

Parameters

A	MxN matrix
b0	Initial vector of length P to start the sequence
mu	Original guess at an eigenvalue
max_iterations	Maximum number of iterations to perform

Returns: Vector v such that $Av\approx\lambda v$ where $\lambda$ is the eigenvalue of A closest to mu

◆ IsCompanion()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsCompanion ( const Matrix< T, M, N, Flags > & A )

Checks if A is a companion matrix.

Parameters

A	MxN matrix

Returns: True if A is a companion matrix

◆ IsDiagonal()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsDiagonal ( const Matrix< T, M, N, Flags > & A )

Checks if A is a diagonal matrix.

Parameters

A	MxN matrix

Returns: True if A is diagonal

◆ IsHermitian()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsHermitian ( const Matrix< T, M, N, Flags > & A )

Checks if a matrix is Hermitian.

Parameters

A	MxN matrix

Returns: True if A is square and $a_{ij}=\overline{a_{ji}}$ for all i and j.

◆ IsHessenberg()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsHessenberg ( const Matrix< T, M, N, Flags > & a )

Checks if a matrix is Hessenberg.

Parameters

A	MxN matrix

Returns: True if A is either lower or upper Hessenberg

◆ IsIdentity()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsIdentity ( const Matrix< T, M, N, Flags > & A )

Checks if A is the identity matrix.

Parameters

A	MxN matrix

Returns: True if A is the identity

◆ IsLowerHessenberg()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsLowerHessenberg ( const Matrix< T, M, N, Flags > & A )

Checks if a matrix is lower Hessenberg.

Parameters

A	MxN matrix

Returns: True if A is square and $a_{ij}=0$ for all i,j such that $j>i+1$

◆ IsLowerTriangular()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsLowerTriangular ( const Matrix< T, M, N, Flags > & A )

Checks if A is lower triangular form.

Parameters

A	MxN matrix

Returns: True if A is lower triangular form

◆ IsReal() [1/2]

template<typename T >

bool Linear::IsReal ( const Complex< T > & z )

Checks if a complex number $z=a+bi$ is real.

Parameters

z	Complex number.

Returns: Returns true if $b=0$.

◆ IsReal() [2/2]

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsReal ( const Matrix< T, M, N, Flags > & A )

Determines if a matrix is real or complex.

Parameters

A	MxN Matrix

Returns: Returns true if every entry of A is real.

◆ IsSquare()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsSquare ( Matrix< T, M, N, Flags > A )

Checks if the matrix is squre.

Parameters

A	MxN matrix

Returns: True if M=N.

◆ IsSymmetric()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsSymmetric ( const Matrix< T, M, N, Flags > & A )

Checks if a matrix is symmetric.

Parameters

A	MxN matrix

Returns: True if A is square and $a_{ij}=a_{ji}$ for all i and j.

◆ IsTriangular()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsTriangular ( const Matrix< T, M, N, Flags > & A )

Checks if A is lower triangular form or upper triangular form.

Parameters

A	MxN matrix

Returns: True if A is upper triangular form or lower triangular form

◆ IsTridiagonal()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsTridiagonal ( const Matrix< T, M, N, Flags > & a )

Checks if a matrix is Tridiagonal.

Parameters

A	MxN matrix

Returns: True if A is both lower and upper Hessenberg

◆ IsUpperHessenberg()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsUpperHessenberg ( const Matrix< T, M, N, Flags > & A )

Checks if a matrix is upper Hessenberg.

Parameters

A	MxN matrix

Returns: True if A is square and $a_{ij}=0$ for all i,j such that $i>j+1$

◆ IsUpperTriangular()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsUpperTriangular ( const Matrix< T, M, N, Flags > & A )

Checks if A is upper triangular form.

Parameters

A	MxN matrix

Returns: True if A is upper triangular form

◆ IsVector()

template<typename T , size_t M, size_t N, unsigned int Flags>

bool Linear::IsVector ( const Matrix< T, M, N, Flags > & A )

Checks if a matrix is a row or column vector.

Parameters

A	MxN matrix

Returns: True if M=1 or N=1

◆ Kronecker()

template<typename T , size_t M, size_t N, unsigned int Flags1, size_t P, size_t Q, unsigned int Flags2>

Matrix<T,MP,QN,Flags1> Linear::Kronecker	(	const Matrix< T, M, N, Flags1 > &	A,
		const Matrix< T, P, Q, Flags2 > &	B
	)

Computes the (MP)x(NQ) block matrix.

\[ A\otimes B = \begin{bmatrix} a_{0,0}B & \dots & a_{0,N-1}B \\ \vdots & \ddots & \vdots \\ a_{M-1,0}B & \dots & a_{M-1,N-1}B \end{bmatrix}. \]

Parameters

A	MxN Matrix
B	PxQ Matrix

Returns: (MP)x(NQ) Matrix

◆ KroneckerSum()

template<typename T , size_t M, size_t N, unsigned int Flags1, size_t P, size_t Q, unsigned int Flags2>

SquareMatrix<T,N*Q,Flags1> Linear::KroneckerSum	(	const Matrix< T, M, N, Flags1 > &	A,
		const Matrix< T, P, Q, Flags2 > &	B
	)

Computes the (NQ)x(NQ) Kronecker sum defined by.

\[ A \oplus B = A\otimes I_Q + I_N\otimes B \]

where $I_N$ represents the NxN identity matrix.

Parameters

A	NxN Matrix
B	QxQ Matrix

Returns: (NQ)x(NQ) Matrix

◆ Log() [1/9]

template<typename T >

Complex<T> Linear::Log ( const Complex< T > & z )

Computes the principal natural log of a complex number $z$.

If we use Euler's formula and rewrite $z=re^{\varphi i}$, then $\log z=\log r+\varphi i$. Generally this would be considered a multivalued function as $\log z=\{\log r+(\varphi+2\pi k)i\mid k\in\mathbb{Z}\}$. But we just take the principal natural log, i.e., set $k=0$.

Parameters

z	Complex number

Returns: Complex number.

◆ Log() [2/9]

template<typename T >

Complex<T> Linear::Log	(	const Complex< T > &	z,
		const Complex< T > &	base
	)

Computes the $\log_{base}z$.

This is done via change-of-base formula, $\log_{base}z=\log z/\log base$.

Parameters

z	Complex number.
base	Complex number.

Returns: Complex number.

◆ Log() [3/9]

template<typename T >

Complex<T> Linear::Log	(	const Complex< T > &	z,
		T	base
	)

Computes the $\log_{base}z$.

This is done via change-of-base formula, $\log_{base}z=\log z/\log base$.

Parameters

z	Complex number.
base	Real number.

Returns: Complex number.

◆ Log() [4/9]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Log ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix $B$ defined by $b_{ij}=\log a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Log() [5/9]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Log	(	Matrix< T, M, N, Flags >	A,
		Complex< T >	base
	)

Computes the MxN Matrix $B$ defined by $b_{ij}=\log_{base} a_{ij}$.

Parameters

A	MxN Matrix
base	Complex number

Returns: MxN Matrix

◆ Log() [6/9]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Log	(	Matrix< T, M, N, Flags >	A,
		T	base
	)

Computes the MxN Matrix $B$ defined by $b_{ij}=\log_{base} a_{ij}$.

Parameters

A	MxN Matrix
base	Real number

Returns: MxN Matrix

◆ Log() [7/9]

template<typename T >

T Linear::Log ( T x )

Computes the natural log of a real number.

Parameters

x	Real number.

Returns: Real number.

◆ Log() [8/9]

template<typename T >

Complex<T> Linear::Log	(	T	x,
		const Complex< T > &	base
	)

Computes the $\log_{base}x$.

This is done via change-of-base formula, $\log_{base}x=\log x/\log base$.

Parameters

x	Real number.
base	Complex number.

Returns: Complex number.

◆ Log() [9/9]

template<typename T >

T Linear::Log	(	T	x,
		T	base
	)

Computes the $\log_{base}x$.

This is done via change-of-base formula, $\log_{base}x=\log x/\log base$.

Parameters

x	Real number.
base	Real number.

Returns: Real number.

◆ LookAt()

template<typename T , size_t N1, size_t N2, size_t N3, size_t N4, unsigned int Flags = 0>

std::enable_if<((N1==3\|\|N1==Dynamic)\|\|(N2==3\|\|N2==Dynamic)\|\|(N3==3\|\|N3==Dynamic)\|\|(N4==3\|\|N4==Dynamic)), SquareMatrix<T,4,Flags> >::type Linear::LookAt	(	Vector< T, N1 >	right,
		Vector< T, N2 >	up,
		Vector< T, N3 >	dir,
		Vector< T, N4 >	eye
	)

Computes the 4x4 look at matrix.

Parameters

Flags	Flags to pass to the matrix (default = row major)
right	Right direction
up	Up direction
dir	Forward direction
eye	Camera position

Returns: 4x4 Matrix

◆ MaxNorm()

template<typename T , size_t M, size_t N, unsigned int Flags>

T Linear::MaxNorm ( const Matrix< T, M, N, Flags > & A )

Computes the max norm $\max_{ij}|a_{ij}|$.

Parameters

A	MxN Matrix

Returns: Real number

◆ Minor()

template<typename T , size_t M, size_t N, unsigned int Flags>

Complex<T> Linear::Minor	(	const Matrix< T, M, N, Flags > &	A,
		size_t	i,
		size_t	j
	)

Computes the (i,j)-minor of a square matrix defined by $\det(A_{ij})$ where $A_{ij}$ is the (N-1)x(N-1) matrix obtained by deleting the ith row and jth column.

If A is not square, an exception is thrown. Likewise if i or j is out-of-bounds an exception is thrown.

Parameters

A	MxN matrix
i	Row index
j	Column index

Returns: Complex number

◆ Mod() [1/5]

template<typename T >

Complex<T> Linear::Mod	(	const Complex< T > &	z,
		const Complex< T > &	w
	)

Returns the complex number $(a\bmod c)+(b\bmod d)i$ where $z=a+bi$ and $w=b+di$.

Parameters

z	Complex number.
w	Complex number.

Returns: Complex number.

◆ Mod() [2/5]

template<typename T >

Complex<T> Linear::Mod	(	const Complex< T > &	z,
		T	y
	)

Returns the complex number $(a\bmod y)+(b\bmod y)i$ where $z=a+bi$.

Parameters

z	Complex number.
y	Real number.

Returns: Complex number.

◆ Mod() [3/5]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Mod	(	Matrix< T, M, N, Flags >	A,
		Complex< T >	z
	)

Computes the MxN Matrix $B$ defined by $b_{ij}=a_{ij}\bmod z$.

Parameters

A	MxN Matrix
z	Complex number

Returns: MxN Matrix

◆ Mod() [4/5]

template<typename T , size_t M, size_t N, unsigned int Flags, typename U >

Matrix<T,M,N,Flags> Linear::Mod	(	Matrix< T, M, N, Flags >	A,
		U	y
	)

Computes the MxN Matrix $B$ defined by $b_{ij}=a_{ij}\bmod y$.

Parameters

A	MxN Matrix
y	Real number

Returns: MxN Matrix

◆ Mod() [5/5]

template<typename T >

T Linear::Mod	(	T	x,
		T	y
	)

Computes the floating-point modulus $x\bmod y$.

Parameters

x	Real number.
y	Real number.

Returns: Real number.

◆ Norm()

template<typename T , size_t M, size_t N, unsigned int Flags>

T Linear::Norm	(	const Matrix< T, M, N, Flags > &	A,
		size_t	p = `2`
	)

If $A$ is a vector, it computes the entrywise vector p-norm $\left(\sum_{i=0}^{N-1}|a_i|^p\right)^{1/p}$.

Otherwise it computes the matrix norm $\|A\|_p$. If p=1, it returns $\max_{j=0,\dots,N-1}\sum_{i=0}^{M-1}|a_{ij}|$. If p=2, it returns the largest singular value of $A$. Otherwise an error is returned.

Parameters

A	MxN Matrix
p	Real number (default = 2)

Returns: Real number

◆ Normalize()

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Normalize ( Matrix< T, M, N, Flags > v )

Normalize a vector, $v=\frac{v}{\|v\|}$.

Example: w is the column vector {1/5, 2/5}

Vector2d v = {1, 2};
Vector2d w = Normalize(v);

Parameters

v	Row/column vector

Returns: Column vector if v is a column vector, otherwise row vector.

◆ Nullity()

template<typename T , size_t M, size_t N, unsigned int Flags>

unsigned int Linear::Nullity ( const Matrix< T, M, N, Flags > & A )

Computes the dimension of the A's null space.

Parameters

A	MxN matrix

Returns: dim(Null(A))

◆ NullSpace()

template<typename T , size_t M, size_t N, unsigned int Flags>

std::vector<Vector<T,N> > Linear::NullSpace ( const Matrix< T, M, N, Flags > & A )

Computes a basis for the null space of A.

The null space of a matrix is the set of vectors v such that Av=0.

Parameters

A	MxN matrix

Returns: List of vectors v such that $Null(A)=span\{v[0],\dots,v[len(v)-1]\}$

◆ One() [1/2]

template<typename T , size_t M, size_t N, unsigned int Flags = 0>

Matrix<T,M,N,Flags> Linear::One ( )

Creates the MxN all ones matrix.

Example: A is the 2x3 matrix {{1,1,1}, {1,1,1}}

Matrix<double,2,3> A = One<double, 2, 3>();

Parameters

T	Type
M	Number of rows
N	Number of columns
Flags	Flags to pass to the matrix (default = row major)

Returns: MxN Matrix

◆ One() [2/2]

template<typename T , unsigned int Flags = 0>

Matrix<T,Dynamic,Dynamic,Flags> Linear::One	(	size_t	nrows,
		size_t	ncols
	)

Dynamically creates the nrowsxncols all ones matrix.

Example: A is the 2x3 matrix {{1,1,1}, {1,1,1}}

MatrixXd A = One<double>(2, 3);

Parameters

T	Type
Flags	Flags to pass to the matrix (default = row major)
nrows	Number of rows
ncols	Number of columns

Returns: nrowsxncols Matrix

◆ Orthographic()

template<typename T , unsigned int Flags = 0>

SquareMatrix<T,4,Flags> Linear::Orthographic	(	T	left,
		T	right,
		T	bottom,
		T	top,
		T	near,
		T	far
	)

Computes the 4x4 orthographic projection matrix.

See https://en.wikipedia.org/wiki/Orthographic_projection for more information.

Parameters

Flags Flags to pass to the matrix (default = row major)

Returns: 4x4 Matrix

◆ Perspective()

template<typename T , unsigned int Flags = 0>

SquareMatrix<T,4,Flags> Linear::Perspective	(	T	fov,
		T	aspect,
		T	near,
		T	far
	)

Computes the 4x4 perspective projection matrix.

Parameters

Flags	Flags to pass to the matrix (default = row major)
fov	Field of view in radians
aspect	Aspect ratio
near	Distance to near plane
far	Distance to far plane

Returns: 4x4 Matrix

◆ Pow() [1/4]

template<typename T >

Complex<T> Linear::Pow	(	const Complex< T > &	z,
		const Complex< T > &	w
	)

Computes the complex number $z^w$.

This is done using the exponential, $z^w:=e^{w\log z}$. Again this is a multivalued function but we take the principal value.

Parameters

z	Complex number.
w	Complex number.

Returns: Complex number.

◆ Pow() [2/4]

template<typename T , size_t M, size_t N, unsigned int Flags, typename U >

Matrix<T,M,N,Flags> Linear::Pow	(	const Matrix< T, M, N, Flags > &	A,
		U	power
	)

◆ Pow() [3/4]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Pow	(	Matrix< T, M, N, Flags >	A,
		Complex< T >	power
	)

If $A$ is not square, it computes the MxN Matrix $B$ defined by $b_{ij}=a_{ij}^{power}$.

Otherwise it attempts to computes the matrix power $A^{power}$. If $A=VDV^{-1}$ with $D=diag(d_1,\dots,d_N)$ a diagonal matrix, then $A^{power}=VD^{power}V^{-1}$ and $D^{power}=diag(d_1^{power},\dots,d_N^{power})$. If no such decomposition can be found and power is an integer, it multiplies A by itself |power|-times then taking the inverse if power<0. Otherwise it gives up and throws an exception.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Pow() [4/4]

template<typename T >

T Linear::Pow	(	T	x,
		T	y
	)

Computes the real number $x^y$.

Note this will not safeguard expressions such as $(-1)^{1/2}$.

Parameters

x	Real number.
y	Real number.

Returns: Real number.

◆ PowerIteration()

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P>

std::enable_if<(P==N\|\|P==Dynamic\|\|N==Dynamic), Eigenpair<T,P> >::type Linear::PowerIteration	(	const Matrix< T, M, N, Flags > &	A,
		Vector< T, P >	b0,
		unsigned int	max_iterations
	)

Performs power iteration on A.

Power iteration, defined by the sequence $b_{k+1}=\frac{Ab_k}{\|Ab_k\|}$ is a simple algorithm that computes the greatest (in magnitude) eigenvalue of A along with it's corresponding eigenvector. Convergence is often slow.

More information: https://en.wikipedia.org/wiki/Power_iteration

Parameters

A	MxN matrix
b0	Initial vector of length P to start the sequence.
max_iterations	Maximum number of iterations to perform

Returns: Pair $(\lambda,v)$ such that $Av\approx \lambda v$

◆ Proj()

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P, size_t Q, unsigned int Flags2>

Matrix<T,P,Q,Flags2> Linear::Proj	(	const Matrix< T, M, N, Flags > &	v,
		const Matrix< T, P, Q, Flags2 > &	onto
	)

Computes the vector projection $\proj_{onto}v = \frac{Dot(onto,v)}{Dot(onto,onto)}onto$.

If onto is the zero vector, by definition this function will return the zero vector. If either v or onto is not a vector, or they have different lengths, an exception is thrown.

Example: w is the column vector {2,0}

Vector2d v = {2, 2};
Vector2d onto = {1, 0};
Vector2d w = Proj(v,onto);

Parameters

v	Row/column vector
onto	Row/column vector

Returns: Column vector if onto is a column vector, otherwise row vector.

◆ Random() [1/2]

template<typename T , size_t M, size_t N, unsigned int Flags = 0>

Matrix<T,M,N,Flags> Linear::Random	(	Complex< T >	min = `Complex<T>(0.0)`,
		Complex< T >	max = `Complex<T>(1.0)`
	)

Creates an MxN random matrix $A$ where all entries are randomly determined between min and max.

If $min=a+bi$ and $max=c+di$, then each entry of $A$ will be of the form $x+iy$ where $a\le x\le c$ and $b\le y\le d$.

Parameters

T	Type
M	Number of rows
N	Number of columns
Flags	Flags to pass to the matrix (default = row major)
min	Complex number (default = 0)
max	Complex number (default = 1+0i)

Returns: MxN Matrix

◆ Random() [2/2]

template<typename T , unsigned int Flags = 0>

Matrix<T,Dynamic,Dynamic,Flags> Linear::Random	(	size_t	nrows,
		size_t	ncols,
		Complex< T >	min = `Complex<T>(0.0)`,
		Complex< T >	max = `Complex<T>(1.0)`
	)

Dynamically creates an nrowsxncols random matrix $A$ where all entries are randomly determined between min and max.

If $min=a+bi$ and $max=c+di$, then each entry of $A$ will be of the form $x+iy$ where $a\le x\le c$ and $b\le y\le d$.

Parameters

T	Type
Flags	Flags to pass to the matrix (default = row major)
nrows	Number of rows
ncols	Number of columns
min	Complex number (default = 0)
max	Complex number (default = 1+0i)

Returns: nrowsxncols Matrix

◆ Rank()

template<typename T , size_t M, size_t N, unsigned int Flags>

unsigned int Linear::Rank ( const Matrix< T, M, N, Flags > & A )

Computes the dimension of the A's column space.

Parameters

A	MxN matrix

Returns: dim(colsp(A))

◆ RemoveColumn()

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,(N==Dynamic?Dynamic:N-1),Flags> Linear::RemoveColumn	(	Matrix< T, M, N, Flags >	A,
		size_t	i
	)

Returns the Mx(N-1) submatrix formed by removing the ith column.

Parameters

A	MxN matrix
i	Column to remove

Returns: Mx(N-1) submatrix of A

◆ RemoveRow()

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,(M==Dynamic?Dynamic:M-1),N,Flags> Linear::RemoveRow	(	Matrix< T, M, N, Flags >	A,
		size_t	i
	)

Returns the (M-1)xN submatrix formed by removing the ith row.

Parameters

A	MxN matrix
i	Row to remove

Returns: (M-1)xN submatrix of A

◆ RemoveRowAndColumn()

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,(M==Dynamic?Dynamic:M-1),(N==Dynamic?Dynamic:N-1),Flags> Linear::RemoveRowAndColumn	(	Matrix< T, M, N, Flags >	A,
		size_t	i,
		size_t	j
	)

Returns the (M-1)x(N-1) submatrix formed by removing the ith row and jth column.

Parameters

A	MxN matrix
i	Row to remove
j	Column to remove

Returns: (M-1)x(N-1) submatrix of A

◆ Rotation() [1/2]

template<size_t P, typename T , size_t N = 4, unsigned int Flags = 0>

std::enable_if<((N==3\|\|N==4)&&(P==3\|\|P==Dynamic)), SquareMatrix<T,N,Flags> >::type Linear::Rotation	(	T	theta,
		Vector< T, P >	axis
	)

Computes the 4x4 rotatation matrix about an axis.

If N=3, it removes the last row and column.

Parameters

N	Size of matrix (either 3 or 4, default = 4)
Flags	Flags to pass to the matrix (default = row major)
theta	Angle to use in radians
axis	Vector of length 3 representing the axis of rotation

Returns: NxN Matrix

◆ Rotation() [2/2]

template<size_t P, typename T , size_t N = 4, unsigned int Flags = 0>

std::enable_if<((N==3||N==4)&&(P==4||P==Dynamic)), SquareMatrix<T,N,Flags> >::type Linear::Rotation ( Vector< T, P > q )

Computes the 4x4 rotatation matrix given a quaternion q.

If N=3, it removes the last row and column.

Parameters

N	Size of matrix (either 3 or 4, default = 4)
Flags	Flags to pass to the matrix (default = row major)
q	Vector of length 4 representing quaternion q[0]+q[1]i+q[2]j+q[3]k

Returns: NxN Matrix

◆ RotationX()

template<typename T , size_t N = 4, unsigned int Flags = 0>

std::enable_if<(N==3||N==4), SquareMatrix<T,N,Flags> >::type Linear::RotationX ( T theta )

Computes the 4x4 rotatation matrix.

\[ \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta & 0 \\ 0 & \sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}. \]

If N=3, it removes the last row and column.

Parameters

N	Size of matrix (either 3 or 4, default = 4)
Flags	Flags to pass to the matrix (default = row major)
theta	Angle to use in radians

Returns: NxN Matrix

◆ RotationY()

template<typename T , size_t N = 4, unsigned int Flags = 0>

std::enable_if<(N==3||N==4), SquareMatrix<T,N,Flags> >::type Linear::RotationY ( T theta )

Computes the 4x4 rotatation matrix.

\[ \begin{bmatrix} \cos\theta & 0 & \sin\theta & 0 \\ 0 & 1 & 0 & 0 \\ -\sin\theta & 0 & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}. \]

If N=3, it removes the last row and column.

Parameters

N	Size of matrix (either 3 or 4, default = 4)
Flags	Flags to pass to the matrix (default = row major)
theta	Angle to use in radians

Returns: NxN Matrix

◆ RotationZ()

template<typename T , size_t N, unsigned int Flags = 0>

std::enable_if<(N==3||N==4), SquareMatrix<T,N,Flags> >::type Linear::RotationZ ( T theta )

Computes the 4x4 rotatation matrix.

\[ \begin{bmatrix} \cos\theta & -\sin\theta & 0 & 0 \\ \sin\theta & \cos\theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}. \]

If N=3, it removes the last row and column.

Parameters

N	Size of matrix (either 3 or 4, default = 4)
Flags	Flags to pass to the matrix (default = row major)
theta	Angle to use in radians

Returns: NxN Matrix

◆ Round() [1/2]

template<typename T >

Complex<T> Linear::Round ( const Complex< T > & z )

Rounds the real and imaginary parts of $z$ to the nearest integer.

Parameters

z	Complex number.

Returns: Complex number.

◆ Round() [2/2]

template<typename T >

T Linear::Round ( T x )

Rounds $x$ to the nearest integer.

Parameters

x	Real number.

Returns: Real number.

◆ RowAugmented()

template<typename T , size_t M1, size_t N1, unsigned int Flags1, size_t M2, size_t N2, unsigned int Flags2>

std::enable_if<(N1==N2\|\|N1==Dynamic\|\|N2==Dynamic), Matrix<T,(M1==Dynamic\|\|M2==Dynamic?Dynamic:M1+M2),N1,Flags1> >::type Linear::RowAugmented	(	const Matrix< T, M1, N1, Flags1 > &	top,
		const Matrix< T, M2, N2, Flags2 > &	bottom
	)

Creates the (M1+M2)xN1 row augmented matrix.

\[ \begin{bmatrix} top \\ \hline bottom \end{bmatrix}. \]

If the two matrices have differing numbers of columns, an exception is thrown.

Parameters

top	M1xN1 Matrix
bottom	M2xN2 Matrix

Returns: (M1+M2)xN1 Matrix

◆ RREF()

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::RREF ( Matrix< T, M, N, Flags > A )

Computes A's reduced row echelon form.

Parameters

A	MxN matrix

Returns: MxN matrix

◆ Scaling()

template<typename T , size_t N, unsigned int Flags = 0>

SquareMatrix<T,N,Flags> Linear::Scaling ( const Vector< T, N > & s )

Computes the NxN scaling matrix S such that.

\[ Sx = \begin{bmatrix} s_0 & \dots & \\ & \ddots & \\ & & s_{N-1} \end{bmatrix} \begin{bmatrix} x_0\\ \vdots \\ x_{N-1}\end{bmatrix} = \begin{bmatrix} s_0x_0\\ \vdots \\ s_{N-1}x_{N-1} \end{bmatrix}. \]

Parameters

Flags	Flags to pass to the matrix (default = row major)
s	Vector of length N

Returns: NxN Matrix

◆ Sec() [1/3]

template<typename T >

Complex<T> Linear::Sec ( const Complex< T > & z )

Returns the complex number $\sec z=\frac{2}{e^{iz}+e^{-iz}}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Sec() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Sec ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\sec a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Sec() [3/3]

template<typename T >

T Linear::Sec ( T x )

Returns the real number $\sec x=1/\cos x$.

Parameters

x	Real number.

Returns: Real number.

◆ Sech() [1/3]

template<typename T >

Complex<T> Linear::Sech ( const Complex< T > & z )

Returns the complex number $sech z=\frac{2}{e^z+e^{-z}}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Sech() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Sech ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=sech a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Sech() [3/3]

template<typename T >

T Linear::Sech ( T x )

Returns the real number $sech x = 1/\cosh x$.

Parameters

x	Real number.

Returns: Real number.

◆ SeedRandom()

void Linear::SeedRandom ( unsigned int seed )

Seeds the random number generator used in Random.

Parameters

seed	Seed to feed random_number_generator

◆ Sign() [1/3]

template<typename T >

Complex<T> Linear::Sign ( const Complex< T > & z )

Takes a complex number $z$ and computes $sgn(z)$.

If $z$ is real and if $z>0$ it returns $1$. If $z$ is real and $z=0$ it returns $0$. Otherwise if $z$ is real it returns $-1$. If $z$ is not real, it returns $\frac{z}{|z|}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Sign() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Sign ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix $B$ defined by $b_{ij}=sgn(a_{ij})$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Sign() [3/3]

template<typename T >

T Linear::Sign ( T x )

Takes a real number $x$ and computes $sgn(x)$.

If $x>0$ it returns $1$. If $x=0$ it returns $0$. Otherwise it returns $-1$.

Parameters

x	Real number.

Returns: -1, 0 or 1.

◆ Sin() [1/3]

template<typename T >

Complex<T> Linear::Sin ( const Complex< T > & z )

Returns the complex number $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Sin() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Sin ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\sin a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Sin() [3/3]

template<typename T >

T Linear::Sin ( T x )

Returns the real number $\sin x$.

Parameters

x	Real number.

Returns: Real number.

◆ Sinh() [1/3]

template<typename T >

Complex<T> Linear::Sinh ( const Complex< T > & z )

Returns the complex number $\sinh z=\frac{e^z-e^{-z}}{2}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Sinh() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Sinh ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\sinh a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Sinh() [3/3]

template<typename T >

T Linear::Sinh ( T x )

Returns the real number $\sinh x$.

Parameters

x	Real number.

Returns: Real number.

◆ Solve() [1/2]

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P>

std::enable_if<(M==P\|\|P==Dynamic\|\|M==Dynamic), RowVector<T,M> >::type Linear::Solve	(	const Matrix< T, M, N, Flags > &	A,
		const RowVector< T, P > &	b
	)

Solves the matrix equation xA=b for x.

This just recycles the old solver for Ax=b since $(xA)^\top=A^\top x^\top=b^\top$. If N != P, an exception is thrown. If there is no solution, an exception is thrown.

Parameters

A	MxN matrix
b	Row vector of length P

Returns: Row vector of length M

◆ Solve() [2/2]

template<typename T , size_t M, size_t N, unsigned int Flags, size_t P>

std::enable_if<(M==P\|\|P==Dynamic\|\|M==Dynamic), Vector<T,N> >::type Linear::Solve	(	const Matrix< T, M, N, Flags > &	A,
		const Vector< T, P > &	b
	)

Solves the matrix equation Ax=b for x.

This functions breaks into various cases.

If A is square we have 2 special cases. a. If A is invertible, then $x=A^{-1}b$. b. If A is upper triangular or lower triangular, then we can use forward/backward subsitution to solve.
If those two cases don't hold, we employ Guassian elimination.

If M != P, an exception is thrown. If there is no solution, an exception is thrown.

Parameters

A	MxN matrix
b	Vector of length P

Returns: Vector of length N

◆ Sqrt() [1/3]

template<typename T >

Complex<T> Linear::Sqrt ( const Complex< T > & z )

Takes a complex number $z=a+bi$ and computes it's principal square root $\sqrt{z}=\gamma+\delta i$, where $\gamma=\sqrt{\frac{a+|z|}{2}}$ and $\delta=(sgn b)\sqrt{\frac{-a+|z|}{2}}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Sqrt() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Sqrt ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix $B$ defined by $b_{ij}=\sqrt{a_{ij}}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Sqrt() [3/3]

template<typename T >

Complex<T> Linear::Sqrt ( T x )

Computes the square root of a real number $x$.

If $x<0$ it returns the complex number $\sqrt{|x|}i$.

Parameters

x	Real number.

Returns: Complex number.

◆ SubMatrix() [1/2]

template<size_t P, size_t Q, typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,P,Q,Flags> Linear::SubMatrix	(	Matrix< T, M, N, Flags >	A,
		size_t	i = `0`,
		size_t	j = `0`
	)

Returns the PxQ submatrix located at (i,j) If i+P>M or j+Q > N, an exception is raised.

Example: B is the matrix {{2,3},{5,6}}

Matrix3d A = { {1,2,3}, {4,5,6}, {7,8,9} };
Matrix2d B = SubMatrix<2,2>(A, 0, 1);

Parameters

A	MxN matrix
i	Row offset (default = 0)
j	Column offset (default = 0)

Returns: PxQ submatrix of A

◆ SubMatrix() [2/2]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,Dynamic,Dynamic,Flags> Linear::SubMatrix	(	Matrix< T, M, N, Flags >	A,
		size_t	nrows,
		size_t	ncols,
		size_t	i = `0`,
		size_t	j = `0`
	)

Returns the nrowsxncols submatrix located at (i,j) If i+nrows>M or j+ncols > N, an exception is raised.

Example: B is the matrix {{2,3},{5,6}}

Matrix3d A = { {1,2,3}, {4,5,6}, {7,8,9} };
MatrixXd B = SubMatrix(A, 2, 2, 0, 1);

Parameters

A	MxN matrix
nrows	Number of rows in submatrix
ncols	Number of columns in submatrix
i	Row offset (default = 0)
j	Column offset (default = 0)

Returns: nrowsxncols submatrix of A

◆ SubVector() [1/6]

template<size_t P, typename T , size_t N>

RowVector<T,P> Linear::SubVector	(	const RowVector< T, N > &	v,
		size_t	off = `0`
	)

Constructs a subvector of a row vector.

If P<1 or off+P>N, an exception is thrown.

Example: w is row vector {2,3}

RowVector4d v = { 1, 2, 3, 4 };
RowVector2d w = SubVector<2>(v, 1);

Parameters

P	Length for subvector
v	Row vector of size N
off	Offset to start the subvector (default = 0)

Returns: Row vector of size P

◆ SubVector() [2/6]

template<typename T , size_t N>

RowVector<T,Dynamic> Linear::SubVector	(	const RowVector< T, N > &	v,
		size_t	size,
		size_t	off = `0`
	)

Constructs a subvector of a row vector.

If size<1 or off+size>N, an exception is thrown.

Example: w is dynamic row vector {2,3}

RowVector4d v = { 1, 2, 3, 4 };
RowVectorXd w = SubVector<2>(v, 2, 1);

Parameters

v	Row vector of size N
size	Length for subvector
off	Offset to start the subvector (default = 0)

Returns: Row vector of size size

◆ SubVector() [3/6]

template<size_t P, typename T >

Vector<T,P> Linear::SubVector	(	const Vector< T, 1 > &	v,
		size_t	off = `0`
	)

Handles subvector in the size 1 case.

If P is not one or off is not zero, an exception is raised.

Example: w is vector {1}

Vector<double,1> v = { 1 };
Vector<double,1> w = SubVector<1>(v);

Parameters

P	Length for subvector
v	Vector of length 1
off	Offset to start the subvector (default = 0)

Returns: v

◆ SubVector() [4/6]

template<typename T >

Vector<T,Dynamic> Linear::SubVector	(	const Vector< T, 1 > &	v,
		size_t	size,
		size_t	off = `0`
	)

Handles subvector in the size 1 case.

If size is not one or off is not zero, an exception is raised.

Example: w is dynamic vector {1}

Vector<double,1> v = { 1 };
VectorXd w = SubVector(v, 1);

Parameters

v	Row vector of size 1
size	Size for subvector
off	Offset to start the subvector (default = 0)

Returns: v

◆ SubVector() [5/6]

template<size_t P, typename T , size_t N>

Vector<T,P> Linear::SubVector	(	const Vector< T, N > &	v,
		size_t	off = `0`
	)

Constructs a subvector of a column vector.

If P<1 or off+P>N, an exception is thrown.

Example: w is column vector {2,3}

Vector4d v = { 1, 2, 3, 4 };
Vector2d w = SubVector<2>(v, 1);

Parameters

P	Length for subvector
v	Column vector of size N
off	Offset to start the subvector (default = 0)

Returns: Column vector of size P

◆ SubVector() [6/6]

template<typename T , size_t N>

Vector<T,Dynamic> Linear::SubVector	(	const Vector< T, N > &	v,
		size_t	size,
		size_t	off = `0`
	)

Constructs a subvector of a column vector.

If size<1 or off+size>N, an exception is thrown.

Example: w is dynamic column vector {2,3}

Vector4d v = { 1, 2, 3, 4 };
VectorXd w = SubVector(v, 2, 1);

Parameters

v	Column vector of size N
size	Length for subvector
off	Offset to start the subvector (default 0)

Returns: Column vector of size size

◆ Tan() [1/3]

template<typename T >

Complex<T> Linear::Tan ( const Complex< T > & z )

Returns the complex number $\tan z=\frac{1}{i}\left(\frac{e^{2iz}-1}{e^{2iz}+1}\right)$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Tan() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Tan ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\tan a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Tan() [3/3]

template<typename T >

T Linear::Tan ( T x )

Returns the real number $\tan x$.

Parameters

x	Real number.

Returns: Real number.

◆ Tanh() [1/3]

template<typename T >

Complex<T> Linear::Tanh ( const Complex< T > & z )

Returns the complex number $\tanh z=\frac{e^z-e^{-z}}{e^z+e^{-z}}$.

Parameters

z	Complex number.

Returns: Complex number.

◆ Tanh() [2/3]

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,M,N,Flags> Linear::Tanh ( Matrix< T, M, N, Flags > A )

Computes the MxN Matrix$B$ defined by $b_{ij}=\tanh a_{ij}$.

Parameters

A	MxN Matrix

Returns: MxN Matrix

◆ Tanh() [3/3]

template<typename T >

T Linear::Tanh ( T x )

Returns the real number $\tanh x$.

Parameters

x	Real number.

Returns: Real number.

◆ Trace()

template<typename T , size_t M, size_t N, unsigned int Flags>

Complex<T> Linear::Trace ( const Matrix< T, M, N, Flags > & A )

Computes the trace of a square matrix $\sum_{i=0}^{N-1}a_{ii}$.

If A is not square, an exception is thrown.

Parameters

A	MxN matrix

Returns: Complex number

◆ Translation()

template<typename T , size_t N, unsigned int Flags = 0>

SquareMatrix<T,(N==Dynamic?Dynamic:N+1),Flags> Linear::Translation ( const Vector< T, N > & t )

Computes the (N+1)x(N+1) translation matrix T such that.

\[ Tx = \begin{bmatrix} 1 & 0 & \dots & 0 & t_0 \\ 0 & 1 & & 0 & t_1 \\ \vdots & & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & 1 & t_{N-1} \\ 0 & 0 & \dots & 0 & 1 \end{bmatrix} \begin{bmatrix} x_0\\ x_1 \\ \vdots \\ x_{N-1} \\ 1\end{bmatrix} = \begin{bmatrix} x_0+t_0\\ x_1+t_1\\ \vdots \\ x_{N-1}+t_{N-1} \\ 1 \end{bmatrix}. \]

Parameters

Flags	Flags to pass to the matrix (default = row major)
t	Vector of length N

Returns: (N+1)x(N+1) Matrix

◆ Transpose()

template<typename T , size_t M, size_t N, unsigned int Flags>

Matrix<T,N,M,Flags> Linear::Transpose ( const Matrix< T, M, N, Flags > & A )

Returns the NxM matrix $B=A^\top$ defined by $b_{ij}=a_{ji}$.

Parameters

A	MxN matrix

Returns: NxM matrix

◆ VietaFormulaHelper()

template<typename T , size_t N>

void Linear::VietaFormulaHelper	(	size_t	data[],
		size_t	start,
		size_t	end,
		size_t	index,
		size_t	k,
		Complex< T > &	sum,
		const Vector< T, N > &	roots
	)

◆ WielandtDeflationAlgorithm()

template<typename T , size_t M, size_t N, unsigned int Flags>

Vector<T,N> Linear::WielandtDeflationAlgorithm	(	const Matrix< T, M, N, Flags > &	A,
		unsigned int	max_iterations = `100`
	)

Performs a combination of deflation and power iteration to get every eigenpair of A Since power iteration returns the largest eigenvalue, if we want to compute the eigenvalues we need to use deflation.

The general idea is that once an eigenvalue is calculated, we reduce A to a slightly smaller matrix which has the same eigenvalues except removing the one we just calculated. Repeating this process eventually computes all eigenvalues.

Parameters

A	MxN matrix
max_iterations	Maximum number of iterations to perform for each call of PowerIteration

Returns: Column vector of eigenvalues.

◆ Zero() [1/2]

template<typename T , size_t M, size_t N, unsigned int Flags = 0>

Matrix<T,M,N,Flags> Linear::Zero ( )

Creates the MxN all zeros matrix.

Example: A is the 2x3 matrix {{0,0,0}, {0,0,0}}

Matrix<double,2,3> A = Zero<double, 2, 3>();

Parameters

T	Type
M	Number of rows
N	Number of columns
Flags	Flags to pass to the matrix (default = row major)

Returns: MxN Matrix

◆ Zero() [2/2]

template<typename T , unsigned int Flags = 0>

Matrix<T,Dynamic,Dynamic,Flags> Linear::Zero	(	size_t	nrows,
		size_t	ncols
	)

Dynamically creates the nrowsxncols all zeros matrix.

Example: A is the 2x3 matrix {{0,0,0}, {0,0,0}}

MatrixXd A = Zero<double>(2, 3);

Parameters

T	Type
Flags	Flags to pass to the matrix (default = row major)
nrows	Number of rows
ncols	Number of columns

Returns: nrowsxncols Matrix

Variable Documentation

◆ ColumnMajor

const unsigned int Linear::ColumnMajor = 0x0001

◆ Dynamic

const unsigned int Linear::Dynamic = 0

◆ random_number_generator

std::mt19937 Linear::random_number_generator

◆ RowMajor

const unsigned int Linear::RowMajor = 0x0000

◆ Tol

double Linear::Tol = 0.00001

Classes

Typedefs

Enumerations

Functions

Variables

Typedef Documentation

◆ Complexd

◆ Complexf

◆ Complexi

◆ Matrix2d

◆ Matrix2f

◆ Matrix2ld

◆ Matrix2x2d

◆ Matrix2x2f

◆ Matrix2x2ld

◆ Matrix2x3d

◆ Matrix2x3f

◆ Matrix2x3ld

◆ Matrix2x4d

◆ Matrix2x4f

◆ Matrix2x4ld

◆ Matrix3d

◆ Matrix3f

◆ Matrix3ld

◆ Matrix3x2d

◆ Matrix3x2f

◆ Matrix3x2ld

◆ Matrix3x3d

◆ Matrix3x3f

◆ Matrix3x3ld

◆ Matrix3x4d

◆ Matrix3x4f

◆ Matrix3x4ld

◆ Matrix4d

◆ Matrix4f

◆ Matrix4ld

◆ Matrix4x2d

◆ Matrix4x2f

◆ Matrix4x2ld

◆ Matrix4x3d

◆ Matrix4x3f

◆ Matrix4x3ld

◆ Matrix4x4d

◆ Matrix4x4f

◆ Matrix4x4ld

◆ MatrixXd

◆ MatrixXf

◆ MatrixXld

◆ RowVector

◆ RowVector2d

◆ RowVector2f

◆ RowVector2ld

◆ RowVector3d

◆ RowVector3f

◆ RowVector3ld

◆ RowVector4d

◆ RowVector4f

◆ RowVector4ld

◆ RowVectorXd

◆ RowVectorXf

◆ RowVectorXld

◆ SquareMatrix

◆ Vector

◆ Vector2d

◆ Vector2f

◆ Vector2ld

◆ Vector3d

◆ Vector3f

◆ Vector3ld

◆ Vector4d

◆ Vector4f

◆ Vector4ld

◆ VectorXd

◆ VectorXf

◆ VectorXld

Enumeration Type Documentation

◆ SVDType

Function Documentation

◆ Abs() [1/3]

◆ Abs() [2/3]