LinAlg Namespace Reference

The Linear Algebra Implementation/Module/Package. More...

Classes
class	CMatrixType
	The Matrix type. More...
class	CVectorType
	The Vector type. More...
Singular Value Decomposition SVD
These functions perform the Singular Value Decomposition SVD of the MxN matrix A. The SVD is defined by: A=U*S*V^T where: U is a M by M orthogonal matrix V is a N by N orthogonal matrix S is a M by N diaggonal matrix. The values in the diagonal are the singular values Parameters: A  the matrix to perform SVD on Returns: U will be resized if it is does not have the correct dimensions V will be resized if it is does not have the correct dimensions S will be resized if it is does not have the correct dimensions. Exceptions: assert(info==0)  for Lapack. Add a throw statement later. Version: Aug 2001 Author: Henrik Aanæs
void	SVD (const CMatrix &A, CMatrix &U, CVector &s, CMatrix &V)
	SVD of A, where the singular values are returned in a Vector.
void	SVD (const CMatrix &A, CMatrix &U, CMatrix &S, CMatrix &V)
	SVD of A, where the singular values are returned in a 'diagonal' Matrix.
CVector	SVD (const CMatrix &A)
	SVD of A, returning only the singular values in a Vector.
Linear Equations
These functions solve the system of linear equations A*x=b for x, where: A is a N by N matrix b is a N vector x is a N vector There a speceilaized functions for symetric positive definite (SPD) matrices yeilding better performance. These are denote by SPD in there function name. Parameters: A  the NxN square matrix b  the N vector Returns: x will be resized if it is does not have the correct dimensions Exceptions: assert(info==0)  for Lapack. Add a throw statement later. assert(A.Row()==A.Col()).  Add a throw statement later. assert(A.Row()==b.Length()).  Add a throw statement later. Version: Aug 2001 Author: Henrik Aanæs
void	LinearSolve (const CMatrix &A, const CVector &b, CVector &x)
	Solves Ax=b for x.
CVector	LinearSolve (const CMatrix &A, const CVector &b)
	Solves Ax=b for x and returns x.
void	LinearSolveSPD (const CMatrix &A, const CVector &b, CVector &x)
	Solves Ax=b for x, where A is SPD.
CVector	LinearSolveSPD (const CMatrix &A, const CVector &b)
	Solves Ax=b for x and returns x, where A is SPD.
Linear Least Squares
These functions solve the Linear Least Squares problem: min_x ||Ax-b||^2 for x, where: || || denotes the 2-norm A is a M by N matrix. For a well formed M>=N and rank (A)=N. See below. b is a M vector. x is a N vector If the solution is not well formed the algorithm provided will find a solution, x, which is not unique, but which sets the objective function to 0. The reson being that the underlining algorithm works by SVD. Parameters: A  the MxN matrix b  the M vector Returns: x will be resized if it is does not have the correct dimensions Exceptions: assert(info==0)  for Lapack. Add a throw statement later. assert(A.Rows()==b.Length());.  Add a throw statement later. Version: Aug 2001 Author: Henrik Aanæs
void	LinearLSSolve (const CMatrix &A, const CVector &b, CVector &x)
	Solves the Linear Least Squares problem min_x ||Ax=b||^2 for x.
CVector	LinearLSSolve (const CMatrix &A, const CVector &b)
	Solves the Linear Least Squares problem min_x ||Ax=b||^2 for x, and returnes x.
Matrix Inversion
These functions inverts the square matrix A. This matrix A must have full rank. Parameters: A  square matrix Returns: InvA the invers of A for one instance. Exceptions: assert(info==0)  for Lapack. This wil among others happen if A is rank deficient. Add a throw statement later. assert(A.Rows()==A.Cols()).  Add a throw statement later. Version: Aug 2001 Author: Henrik Aanæs
void	Invert (CMatrix &A)
	Invertes the square matrix A. That is here A is altered as opposed to the other Invert functions.
void	Inverted (const CMatrix &A, CMatrix &InvA)
	Returns the inverse of the square matrix A in InvA.
CMatrix	Inverted (const CMatrix &A)
	Returns the inverse of the square matrix A.
QR Factorization
This function returns the QR factorization of A, such that Q*R=A where Q is a orthonormal matrix and R is an upper triangular matrix. However, in the case of A.Col()>A.Row(), the last A.Col-A.Row columns of Q are 'carbage' and as such not part of a orthonormal matrix. Parameters: A  the input matrix Returns: Q an orthonormal matrix. (See above) R an upper triangular matrix. Exceptions: assert(info==0)  for Lapack. This wil among others happen if A is rank deficient. Add a throw statement later. assert(A.Rows()>0  && A.Cols()>0). Add a throw statement later. Version: Aug 2001 Author: Henrik Aanæs
void	QRfact (const CMatrix &A, CMatrix &Q, CMatrix &R)
RQ Factorization
This function returns the RQ factorization of A, such that R*Q=A where Q is a orthonormal matrix and R is an upper triangular matrix. However, in the case of A not beeing a square matrix, there might be some fuck up of Q. Parameters: A  the input matrix Returns: Q an orthonormal matrix. (See above) R an upper triangular matrix. Exceptions: assert(info==0)  for Lapack. This wil among others happen if A is rank deficient. Add a throw statement later. assert(A.Rows()>0  && A.Cols()>0). Add a throw statement later. Version: Aug 2001 Author: Henrik Aanæs
void	RQfact (const CMatrix &A, CMatrix &R, CMatrix &Q)
I/O from MatLab
Reads or writes the type to or from a m-file with a somwhat strict format. MatLab can 'load' the matrix by running the m-file in question, and write to the format via the 'LinAlg.m' function. Since MatLab uses double as the working percision, this is the only value of T where this interface to MatLab works. For other types, this functionality is good for loading, saving and debuging. Todo: write the LinAlg.m fuction. Todo: make a template overloading, of these functions. Author: Henrik Aanæs Version: Aug 2001
void	ToMatlab (const CMatrix &M, const std::string &VarName, const std::string &FileName="c:\\test.m", const bool append=true, const std::string &Comment="")
void	FromMatlab (CMatrix &M, const std::string &VarName, const std::string &FileName="c:\\test.m")
Additional matrix operators
These are operators heavily associated with CMAtrixType, but not included in the class definition it self.
template<class T>
CMatrixType< T >	operator+ (const T &Lhs, const CMatrixType< T > &Rhs)
template<class T>
CMatrixType< T >	operator * (const T &Lhs, const CMatrixType< T > &Rhs)
template<class T>
std::ostream &	operator<< (std::ostream &s, const CMatrixType< T > &A)
template<class T>
std::istream &	operator>> (std::istream &s, CMatrixType< T > &A)
Additional vector operators
These are operators heavily associated with CVectorType, but not included in the class definition it self.
template<class T>
CVectorType< T >	operator+ (const T &Lhs, const CVectorType< T > &Rhs)
template<class T>
CVectorType< T >	operator * (const T &Lhs, const CVectorType< T > &Rhs)
template<class T>
std::ostream &	operator<< (std::ostream &s, const CVectorType< T > &a)
template<class T>
std::istream &	operator>> (std::istream &s, CVectorType< T > &a)
Typedefs
typedef CMatrixType< double >	CMatrix
	The Matrix annotation intended for use.
typedef CVectorType< double >	CVector
	The Vector annotation intended for use.
Functions
void	LinearSolveSym (const CMatrix &A, const CVector &b, CVector &x)

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Detailed Description

The Linear Algebra Implementation/Module/Package.

This is the linear algebra package deal with vector and matrix types plus the basic operation on them. The Vector and Matrix Types are templated such that the precision of a given 'system' can be changen. Though the precision or type in mind is the double. Special types for 2 and 3 vectors (CVec2 and CVec3) are included for improved performance. The more advanced linear algebra functions are supplied by lincage to the LaPack package.

Test:

The functionality in this pacage has if nothing else is mentioned been tested by making small functions utilizing the functions. The reason being that the functions here are rather esay to validate due to their functional simplicity.

A list of improvements in mind is:

• Lapack interface
• Vector and Matrix product and so on, with interface to _BLAS_
• iterator thing
• Memory leak check ? inline ?
• Element functor
• Identity matrixs
• Compile and utilize the newer versions of the Lapack routines.
• Have a nameing convention section in the documentaion
• Integrate the design into the documentation
• Have a get data in the matrix and vector instead of &A[0].
• Have 3x3, 3x4 4x1 and 4x4 specialities.
• Look at casting e.g. vec2i=vec2l.
• Error handling in From/To matrix
• Has error handling in Lapack been made ?

Author:

Henrik Aanæs

Version:

Aug 2001

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Typedef Documentation

typedef CMatrixType<double> LinAlg::CMatrix

The Matrix annotation intended for use.

typedef CVectorType<double> LinAlg::CVector

The Vector annotation intended for use.

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Function Documentation

void Invert (  CMatrix &  A  )

Invertes the square matrix A. That is here A is altered as opposed to the other Invert functions.

CMatrix Inverted (  const CMatrix &  A  )

Returns the inverse of the square matrix A.

void Inverted (  const CMatrix &  A, CMatrix &  InvA )

Returns the inverse of the square matrix A in InvA.

CVector LinearLSSolve (  const CMatrix &  A, const CVector &  b )

Solves the Linear Least Squares problem min_x ||Ax=b||^2 for x, and returnes x.

void LinearLSSolve (  const CMatrix &  A, const CVector &  b, CVector &  x )

Solves the Linear Least Squares problem min_x ||Ax=b||^2 for x.

CVector LinearSolve (  const CMatrix &  A, const CVector &  b )

Solves Ax=b for x and returns x.

void LinearSolve (  const CMatrix &  A, const CVector &  b, CVector &  x )

Solves Ax=b for x.

CVector LinearSolveSPD (  const CMatrix &  A, const CVector &  b )

Solves Ax=b for x and returns x, where A is SPD.

void LinearSolveSPD (  const CMatrix &  A, const CVector &  b, CVector &  x )

Solves Ax=b for x, where A is SPD.

CVector SVD (  const CMatrix &  A  )

SVD of A, returning only the singular values in a Vector.

void SVD (  const CMatrix &  A, CMatrix &  U, CMatrix &  S, CMatrix &  V )

SVD of A, where the singular values are returned in a 'diagonal' Matrix.

void SVD (  const CMatrix &  A, CMatrix &  U, CVector &  s, CMatrix &  V )

SVD of A, where the singular values are returned in a Vector.

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