CH_Matrix_Classes Namespace Reference

Matrix Classes and Linear Algebra. See Matrix Classes (namespace CH_Matrix_Classes) for a quick introduction. More...

Classes
class	Indexmatrix
	Matrix class for integral values of type Integer More...
class	IterativeSolverObject
	Abstract interface to iterative methods for solving Ax=b given by an IterativeSystemObject. More...
class	IterativeSystemObject
	Abstract base class for supplying the system for an iterative solver. More...
class	LanczMaxEig
	A Lanczos method allowing spectral transformation by Chebycheff polynomials and premature termination. More...
class	Lanczos
	Abstract interface to Lanzcos methods for computing a few extremal eigenvalues given via a Lanczosmatrix. More...
class	Lanczosmatrix
	Abstract base class for supplying the input matrix for Lanzcosmethods. More...
class	Lanczpol
	A Lanczos method allowing spectral transformation by Chebycheff polynomials and premature termination. More...
struct	mat_greater_index
	"greater"-routine for sorting indices of value arrays by std::sort More...
struct	mat_less_index
	"less"-routine for sorting indices of value arrays by std::sort More...
class	Matrix
	Matrix class for real values of type Real More...
class	MatrixError
	Such an object is generated and passed to MEmessage(), whenever an error occurs. It holds some output information on the error. More...
class	MEdim
	Such an object is generated and passed to MEmessage() whenever matrix dimensions do not agree for a desired operation. More...
class	Memarray
	A simple memory manager for frequent allocation and deallocation of arrays of roughly the same size. More...
class	Memarrayuser
	All derived classes share a common Memarray memory manager, which is generated with the first user and destructed when the last user is destructed. More...
class	MEmem
	Such an object is generated and passed to MEmessage() whenever a memory allocation fails. More...
class	MErange
	Such an object is generated and passed to MEmessage() whenever some index is out of range. More...
class	MinRes
	MinRes method for solving Ax=b with symmetric (indefinite) matrix A and positive definite preconditioner M1 where the preconditioned system is A'x'=b' with A'=M1^{-.5}AM1^{-.5}, x'=M1^{.5}*x and b'=M1^{-.5}b. System matrix and preconditioner are provided by a CH_Matrix_Classes::IterativeSystemObject. More...
class	PCG
	Preconditioned Conjugate Gradient method for solving Ax=b with (symmetric) positive definite matrix A and positive definite preconditioner M1 where the preconditioned system is A'x'=b' with A'=M1^{-.5}AM1^{-.5}, x'=M1^{.5}*x and b'=M1^{-.5}b. System matrix and preconditioner are provided by a CH_Matrix_Classes::IterativeSystemObject. More...
class	Psqmr
	PSQMR method for solving Ax=b with symmetric matrix A and symmetric preconditioner M=M1*M2 (M1 and M2 regular) where the preconditioned system is A'x'=b' with A'=M1^{-1}AM2^{-1}, x'=M2*x and b'=M1^{-1}b. System matrix and preconditioners are provided by a CH_Matrix_Classes::IterativeSystemObject. More...
class	Range
	allows to specify a range of integral values via (from, to, step) meaning {j=from+i*step:j in[from,to],i in {0,1,2,...}} More...
class	Realrange
	allows to specify a range of real values via (from, to, step,tol) meaning {x=from+i*step:x in(from-tol,to+tol),i in {0,1,2,...}} More...
class	Sparsemat
	Matrix class of sparse matrices with real values of type Real More...
class	Sparsesym
	Matrix class of symmetric matrices with real values of type Real More...
class	Symmatrix
	Matrix class of symmetric matrices with real values of type Real More...

Typedefs
typedef int	Integer
	all integer numbers in calculations and indexing are of this type
typedef double	Real
	all real numbers in calculations are of this type

Enumerations
enum	Mtype {   MTglobalfun, MTindexmatrix, MTmatrix, MTsymmetric,   MTsparse, MTsparsesym }
	serves for specifying the source (matrix class or function) of the error More...
enum	MEcode {   ME_unspec, ME_range, ME_mem, ME_dim,   ME_num, ME_warning }
	serves for specifying the error type. More...

Functions
template<class Val >
void	mat_xea (Integer len, Val *x, const Val a)
	Set x[i]=a for len elements of the array x. More...
template<class Val >
void	mat_xea (Integer len, Val *x, const Integer incx, const Val a)
	Set x[i]=a for len elements of the array x incremented by incx. More...
template<class Val >
void	mat_xey (Integer len, Val *x, const Val *y)
	Copy an array of length len to destination x from source y. More...
template<class Val >
void	mat_xey (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy)
	Copy an array of length len to destination x (increment incx) from source y (increment incy). More...
template<class Val >
void	mat_xmey (Integer len, Val *x, const Val *y)
	Set x[i]=-y[i] for len elements of the arrays x and y. More...
template<class Val >
void	mat_xmey (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy)
	Set x[i]=-y[i] for len elements of the arrays x and y incremented by incx and incy, respectively. More...
template<class Val >
void	mat_xemx (Integer len, Val *x)
	Set x[i]=-x[i] for an array of length len. More...
template<class Val >
void	mat_xemx (Integer len, Val *x, const Integer incx)
	Set x[i]=-x[i] for len elements of an array incremented by incx. More...
template<class Val >
void	mat_xemy (Integer len, Val *x, const Val *y)
	Set x[i]=-y[i] for len elements of the arrays x and y. More...
template<class Val >
void	mat_xemy (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy)
	Set x[i]=-y[i] for len elements of the arrays x and y incremented by incx and incy, respectively. More...
template<class Val >
void	mat_xeya (Integer len, Val *x, const Val *y, const Val a)
	Set x[i]=a*y[i] for len elements of the arrays x and y. More...
template<class Val >
void	mat_xeya (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy, const Val a)
	Set x[i]=a*y[i] for len elements of the arrays x and y incremented by incx and incy, respectively. More...
template<class Val >
void	mat_xpey (Integer len, Val *x, const Val *y)
	Set x[i]+=y[i] for len elements of the arrays x and y. More...
template<class Val >
void	mat_xpey (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy)
	Set x[i]+=y[i] for len elements of the arrays x and y incremented by incx and incy, respectively. More...
template<class Val >
void	mat_xhadey (Integer len, Val *x, const Val *y)
	Set x[i]*=y[i] for len elements of the arrays x and y. More...
template<class Val >
void	mat_xinvhadey (Integer len, Val *x, const Val *y)
	Set x[i]/=y[i] for len elements of the arrays x and y, no zero checking! More...
template<class Val >
void	mat_xhadey (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy)
	Set x[i]*=y[i] for len elements of the arrays x and y incremented by incx and incy, respectively. More...
template<class Val >
void	mat_xinvhadey (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy)
	Set x[i]/=y[i] for len elements of the arrays x and y incremented by incx and incy, respectively, no zero checking! More...
template<class Val >
void	mat_xpeya (Integer len, Val *x, const Val *y, const Val a)
	Set x[i]+=a*y[i] for len elements of the arrays x and y. More...
template<class Val >
void	mat_xpeya (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy, const Val a)
	Set x[i]+=a*y[i] for len elements of the arrays x and y incremented by incx and incy, respectively. More...
template<class Val >
void	mat_xbpeya (Integer len, Val *x, const Val *y, const Val a, const Val b)
	Set x[i]=a*y[i]+b*x[i] for len elements of the arrays x and y. More...
template<class Val >
void	mat_xbpeya (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy, const Val a, const Val b)
	Set x[i]=a*y[i]+b*x[i] for len elements of the arrays x and y incremented by incx and incy, respectively. More...
template<class Val >
void	mat_xpea (Integer len, Val *x, const Val a)
	Set x[i]+=a for len elements of the array x. More...
template<class Val >
void	mat_xpea (Integer len, Val *x, const Integer incx, const Val a)
	Set x[i]+=a for len elements of the array x incremented by incx. More...
template<class Val >
void	mat_xmultea (Integer len, Val *x, const Val a)
	Set x[i]*=a for len elements of the array x. More...
template<class Val >
void	mat_xmultea (Integer len, Val *x, const Integer incx, const Val a)
	Set x[i]*=a for len elements of the array x incremented by incx. More...
template<class Val >
void	mat_xdivea (Integer len, Val *x, const Val a)
	Set x[i]/=a for len elements of the array x. More...
template<class Val >
void	mat_xdivea (Integer len, Val *x, const Integer incx, const Val a)
	Set x[i]/=a for len elements of the array x incremented by incx. More...
template<class Val >
void	mat_xmodea (Integer len, Val *x, const Val a)
	Set x[i]%=a for len elements of the array x. More...
template<class Val >
void	mat_xmodea (Integer len, Val *x, const Integer incx, const Val a)
	Set x[i]%=a for len elements of the array x incremented by incx. More...
template<class Val >
void	mat_xeypz (Integer len, Val *x, const Val *y, const Val *z)
	Set x[i]=y[i]+z[i] for len elements of the arrays x, y and z. More...
template<class Val >
void	mat_xeypz (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy, const Val *z, const Integer incz)
	Set x[i]=y[i]+z[i] for len elements of the arrays x, y and z incremented by incx, incy, and incz, respectively. More...
template<class Val >
void	mat_xeymz (Integer len, Val *x, const Val *y, const Val *z)
	Set x[i]=y[i]-z[i] for len elements of the arrays x, y and z. More...
template<class Val >
void	mat_xeymz (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy, const Val *z, const Integer incz)
	Set x[i]=y[i]-z[i] for len elements of the arrays x, y and z incremented by incx, incy, and incz, respectively. More...
template<class Val >
void	mat_xeyapzb (Integer len, Val *x, const Val *y, const Val *z, const Val a, const Val b)
	Set x[i]=a*y[i]+b*z[i] for len elements of the arrays x, y and z. More...
template<class Val >
void	mat_xeyapzb (Integer len, Val *x, const Integer incx, const Val *y, const Integer incy, const Val *z, const Integer incz, const Val a, const Val b)
	Set x[i]=a*y[i]+b*z[i] for len elements of the arrays x, y and z incremented by incx, incy, and incz, respectively. More...
template<class Val >
Val	mat_ip (Integer len, const Val *x, const Val *y, const Val *d=0)
	return sum(x[i]*y[i]) summing over len elements of the arrays x and y. More...
template<class Val >
Val	mat_ip (Integer len, const Val *x, const Integer incx, const Val *y, const Integer incy, const Val *d=0, const Integer incd=1)
	return sum(x[i]*y[i]) summing over len elements of the arrays x and y incremented by incx and incy, respectively. More...
template<class Val >
Val	mat_ip_dense_sparse (const Integer lenx, const Val *x, Integer leny, const Val *yval, const Integer *yind, const Val *d=0)
	return sum(x[yind[j]]*yval[j]) summing over elements of the dense array x and a sparse array representation of y. More...
template<class Val >
Val	mat_ip_sparse_sparse (const Integer lenx, const Val *xval, const Integer *xind, const Integer leny, const Val *yval, const Integer *yind, const Val *d)
	return sum(xval[i]*yval[j] for i,j with xind[i]==yind[j]) summing over elements of the sparse array representations of x and y. More...
template<class Val >
Val	mat_ip (Integer len, const Val *x)
	return sum(x[i]*x[i]) summing over len elements of the array x. More...
template<class Val >
Val	mat_ip (Integer len, const Val *x, const Integer incx)
	return sum(x[i]*x[i]) summing over len elements of the array x incremented by incx. More...
template<class Val >
Val	mat_sum (Integer len, const Val *x)
	returns sum(x[i]) over len elements of the array x. More...
template<class Val >
Val	mat_sum (Integer len, const Val *x, const Integer incx)
	returns sum(x[i]) over len elements of the array x incremented by incx. More...
template<class Val >
bool	mat_equal (Integer len, const Val *x, const Val *y)
	returns true if the elements of the arrays x and y are exactly equal. More...
template<class Val >
void	mat_swap (Integer len, Val *x, Val *y)
	swap values x[i] and y[i] for len elements of the arrays x and y. More...
template<class Val >
void	mat_swap (Integer len, Val *x, const Integer incx, Val *y, const Integer incy)
	swap values x[i] and y[i] for len elements of the arrays x and y incremented by incx and incy, respectively. More...
int	MEmessage (const MatrixError &)
	displays an error message and terminates via abort() or returns 1 in case of warnings.
template<class Mat1 , class Mat2 >
void	chk_mult (const Mat1 &, const Mat2 &, int=0, int=0)
	CONICBUNDLE_DEBUG being undefined, the template function is removed. Otherwise it would check, whether matrices x and y can be multiplied.
template<class T >
int	mem_provide (Memarray &memarray, long provide, long in_use, long &avail, T *&store)
	provide sufficient memory for an existing array, reallocating and copying the old information upon need, returns 0 upon success, !=0 upon failure. More...
template<class T >
int	mem_provide_init0 (Memarray &memarray, long provide, long &avail, T *&store)
	provide sufficient memory for an existing array, reallocating and copying the old information and initializing the new entries to 0, returns 0 upon success, !=0 upon failure. More...
double	abs (double d)
	absolute value of a double
int	abs (int d)
	absolute value of an int
long	abs (long d)
	absolute value of a long
double	max (double a, double b)
	maximum value of two double variables
double	min (double a, double b)
	minimum value of two double variables
int	max (int a, int b)
	maximum value of two int variables
int	min (int a, int b)
	minimum value of two int variables
long	max (long a, long b)
	maximum value of two long vriables
long	min (long a, long b)
	minimum value of two long variables
void	swap (double &a, double &b)
	swaps two double variables
void	swap (long &a, long &b)
	swaps two long variables
void	swap (int &a, int &b)
	swpas two int variables
void	swap (bool &a, bool &b)
	swpas two bool variables
int	sqr (int a)
	return a*a for int a
long	sqr (long a)
	return a*a for long a
double	sqr (double a)
	return a*a for double a
double	sqrt (int a)
	return sqrt for int a
double	sqrt (long a)
	return sqrt for long a
int	sign (int a)
	return the signum of an int a (1 for a>0,-1 for a<0,0 for a==0)
long	sign (long a)
	return the signum of a long a (1 for a>0,-1 for a<0,0 for a==0)
double	sign (double a)
	return the signum of a double a (1. for a>0.,-1. for a<0.,0. for a==0.)
double	sign (double a, double tol)
	return the signum of a double a with tolerance (1. for a>tol,-1. for a<-tol,0. otherwise)
double	d_sign (double a, double b)
	return a if a and b have the same sign, return -a otherwise
Indexmatrix	diag (const Indexmatrix &A)
	returns a column vector v consisting of the elements v(i)=A(i,i), 0<=i<min(row dimension,column dimension)
void	swap (Indexmatrix &A, Indexmatrix &B)
	swap the content of the two matrices A and B (involves no copying)
Indexmatrix &	xbpeya (Indexmatrix &x, const Indexmatrix &y, Integer alpha=1, Integer beta=0, int ytrans=0)
	returns x= alpha*y+beta*x, where y may be transposed (ytrans=1); if beta==0. then x is initialized to the correct size More...
Indexmatrix &	xeyapzb (Indexmatrix &x, const Indexmatrix &y, const Indexmatrix &z, Integer alpha=1, Integer beta=1)
	returns x= alpha*y+beta*z; x is initialized to the correct size; alpha and beta have default value 1 More...
Indexmatrix &	genmult (const Indexmatrix &A, const Indexmatrix &B, Indexmatrix &C, Integer alpha=1, Integer beta=0, int atrans=0, int btrans=0)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0 then C is initialized to the correct size
Indexmatrix	transpose (const Indexmatrix &A)
	returns an Indexmatrix that is the transpose of A
Indexmatrix	abs (const Indexmatrix &A)
	returns an Indexmatrix B with entries B(i,j)=abs(A(i,j))
Integer	trace (const Indexmatrix &A)
	returns the sum of the diagonal elements A(i,i) over all i
Indexmatrix	sumrows (const Indexmatrix &A)
	returns a row vector holding the sum over all rows, i.e., (1 1 ... 1)*A
Indexmatrix	sumcols (const Indexmatrix &A)
	returns a column vector holding the sum over all columns, i.e., A*(1 1 ... 1)^T
Integer	sum (const Indexmatrix &A)
	returns the sum over all elements of A, i.e., (1 1 ... 1)*A*(1 1 ... 1)^T
Indexmatrix	operator< (const Indexmatrix &A, const Indexmatrix &B)
	returns a matrix having elements (i,j)=Integer(A(i,j)<B(i,j)) for all i,j
Indexmatrix	operator<= (const Indexmatrix &A, const Indexmatrix &B)
	returns a matrix having elements (i,j)=Integer(A(i,j)<=B(i,j)) for all i,j
Indexmatrix	operator== (const Indexmatrix &A, const Indexmatrix &B)
	returns a matrix having elements (i,j)=Integer(A(i,j)==B(i,j)) for all i,j
Indexmatrix	operator!= (const Indexmatrix &A, const Indexmatrix &B)
	returns a matrix having elements (i,j)=Integer(A(i,j)!=B(i,j)) for all i,j
Indexmatrix	operator< (const Indexmatrix &A, Integer d)
	returns a matrix having elements (i,j)=Integer(A(i,j)<d) for all i,j
Indexmatrix	operator> (const Indexmatrix &A, Integer d)
	returns a matrix having elements (i,j)=Integer(A(i,j)>d) for all i,j
Indexmatrix	operator<= (const Indexmatrix &A, Integer d)
	returns a matrix having elements (i,j)=Integer(A(i,j)<=d) for all i,j
Indexmatrix	operator>= (const Indexmatrix &A, Integer d)
	returns a matrix having elements (i,j)=Integer(A(i,j)>=d) for all i,j
Indexmatrix	operator== (const Indexmatrix &A, Integer d)
	returns a matrix having elements (i,j)=Integer(A(i,j)==d) for all i,j
Indexmatrix	operator!= (const Indexmatrix &A, Integer d)
	returns a matrix having elements (i,j)=Integer(A(i,j)!=d) for all i,j
Indexmatrix	minrows (const Indexmatrix &A)
	returns a row vector holding in each column the minimum over all rows in this column
Indexmatrix	mincols (const Indexmatrix &A)
	returns a column vector holding in each row the minimum over all columns in this row
Integer	min (const Indexmatrix &A, Integer *iindex=0, Integer *jindex=0)
	returns the minimum value over all elements of the matrix
Indexmatrix	maxrows (const Indexmatrix &A)
	returns a row vector holding in each column the maximum over all rows in this column
Indexmatrix	maxcols (const Indexmatrix &A)
	returns a column vector holding in each row the maximum over all columns in this row
Integer	max (const Indexmatrix &A, Integer *iindex=0, Integer *jindex=0)
	returns the maximum value over all elements of the matrix
Indexmatrix	sortindex (const Indexmatrix &vec, bool nondecreasing=true)
	returns an Indexmatrix ind so that vec(ind(0))<=vec(ind(1))<=...<=vec(ind(vec.dim()-1)) (vec may be rectangular, set nondecreasing=false for opposite order)
void	sortindex (const Indexmatrix &vec, Indexmatrix &ind, bool nondecreasing=true)
	sets ind so that vec(ind(0))<=vec(ind(1))<=...<=vec(ind(vec.dim()-1)) (vec may be rectangular, set nondecreasing=false for opposite order)
std::ostream &	operator<< (std::ostream &o, const Indexmatrix &A)
	output format (all Integer values): nr nc \n A(1,1) A(1,2) ... A(1,nc) \n A(2,1) ... A(nr,nc) \n
std::istream &	operator>> (std::istream &i, Indexmatrix &A)
	input format (all Integer values): nr nc \n A(1,1) A(1,2) ... A(1,nc) \n A(2,1) ... A(nr,nc) \n
Matrix	diag (const Matrix &A)
	returns a column vector v consisting of the elements v(i)=(*this)(i,i), 0<=i<min(row dimension,column dimension)
Matrix &	xbpeya (Matrix &x, const Matrix &y, Real alpha=1., Real beta=0., int ytrans=0)
	returns x= alpha*y+beta*x, where y may be transposed (ytrans=1); if beta==0. then x is initialized to the correct size
Matrix &	xeyapzb (Matrix &x, const Matrix &y, const Matrix &z, Real alpha=1., Real beta=1.)
	returns x= alpha*y+beta*z; x is initialized to the correct size
Matrix &	genmult (const Matrix &A, const Matrix &B, Matrix &C, Real alpha=1., Real beta=0., int atrans=0, int btrans=0)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size
Matrix	transpose (const Matrix &A)
	returns a transposed matrix of A
std::vector< double > &	assign (std::vector< double > &vec, const Matrix &A)
	interpret A as a vector and copy it to a std::vector<double> which is also returned
Matrix &	genmult (const Symmatrix &A, const Matrix &B, Matrix &C, Real alpha, Real beta, int btrans)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size More...
Matrix &	genmult (const Matrix &A, const Symmatrix &B, Matrix &C, Real alpha, Real beta, int atrans)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size More...
Matrix &	genmult (const Sparsesym &A, const Matrix &B, Matrix &C, Real alpha, Real beta, int btrans)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size More...
Matrix &	genmult (const Matrix &A, const Sparsesym &B, Matrix &C, Real alpha, Real beta, int atrans)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size More...
Matrix &	genmult (const Sparsemat &A, const Matrix &B, Matrix &C, Real alpha, Real beta, int atrans, int btrans)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size
Matrix &	genmult (const Matrix &A, const Sparsemat &B, Matrix &C, Real alpha, Real beta, int atrans, int btrans)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size
Matrix	abs (const Matrix &A)
	returns a matrix with elements (i,j)=abs((*this)(i,j)) for all i,j
Real	trace (const Matrix &A)
	=sum(diag(A)) More...
Real	normDsquared (const Matrix &A, const Matrix &d, int atrans=0, int dinv=0)
	returns trace(A^TDA)=|A|^2_D with D=Diag(d). A may be transposed, D may be inverted but there is no check for division by zero
Matrix	sumrows (const Matrix &A)
	returns a row vector holding the sum over all rows, i.e., (1 1 ... 1)*A
Matrix	sumcols (const Matrix &A)
	returns a column vector holding the sum over all columns, i.e., A*(1 1 ... 1)^T
Real	sum (const Matrix &A)
	returns the sum over all elements of A, i.e., (1 1 ... 1)*A*(1 1 ... 1)^T
Matrix	house (const Matrix &x, Integer i=0, Integer j=0, Real tol=1e-10)
	returns the Householder vector of size A.rowdim() for the subcolumn A(i:A.rowdim(),j)
int	rowhouse (Matrix &A, const Matrix &v, Integer i=0, Integer j=0)
	Housholder pre-multiplication of A with Householder vector v; the first nonzero of v is index i, the multplication is applied to all columns of A with index >=j; always returns 0.
int	colhouse (Matrix &A, const Matrix &v, Integer i=0, Integer j=0)
	Housholder post-multiplication of A with Householder vector v; the first nonzero of v is index i, the multplication is applied to all rows of A with index >=j; always returns 0.
Matrix	operator< (const Matrix &A, const Matrix &B)
	returns a matrix having elements (i,j)=Real(A(i,j)<B(i,j)) for all i,j
Matrix	operator<= (const Matrix &A, const Matrix &B)
	returns a matrix having elements (i,j)=Real(A(i,j)<=B(i,j)) for all i,j
Matrix	operator== (const Matrix &A, const Matrix &B)
	returns a matrix having elements (i,j)=Real(A(i,j)==B(i,j)) for all i,j
Matrix	operator!= (const Matrix &A, const Matrix &B)
	returns a matrix having elements (i,j)=Real(A(i,j)!=B(i,j)) for all i,j
Matrix	operator< (const Matrix &A, Real d)
	returns a matrix having elements (i,j)=Real(A(i,j)<d) for all i,j
Matrix	operator> (const Matrix &A, Real d)
	returns a matrix having elements (i,j)=Real(A(i,j)>d) for all i,j
Matrix	operator<= (const Matrix &A, Real d)
	returns a matrix having elements (i,j)=Real(A(i,j)<=d) for all i,j
Matrix	operator>= (const Matrix &A, Real d)
	returns a matrix having elements (i,j)=Real(A(i,j)>=d) for all i,j
Matrix	operator== (const Matrix &A, Real d)
	returns a matrix having elements (i,j)=Real(A(i,j)==d) for all i,j
Matrix	operator!= (const Matrix &A, Real d)
	returns a matrix having elements (i,j)=Real(A(i,j)!=d) for all i,j
Matrix	minrows (const Matrix &A)
	returns a row vector holding in each column the minimum over all rows in this column
Matrix	mincols (const Matrix &A)
	returns a column vector holding in each row the minimum over all columns in this row
Real	min (const Matrix &A, Integer *iindex=0, Integer *jindex=0)
	returns the minimum value over all elements of the matrix
Matrix	maxrows (const Matrix &A)
	returns a row vector holding in each column the maximum over all rows in this column
Matrix	maxcols (const Matrix &A)
	returns a column vector holding in each row the maximum over all columns in this row
Real	max (const Matrix &A, Integer *iindex=0, Integer *jindex=0)
	returns the maximum value over all elements of the matrix
Indexmatrix	sortindex (const Matrix &vec, bool nondecreasing=true)
	returns an Indexmatrix ind so that vec(ind(0))<=vec(ind(1))<=...<=vec(ind(vec.dim()-1)) (vec may be rectangular, set nondecreasing=false for opposite order)
void	sortindex (const Matrix &vec, Indexmatrix &ind, bool nondecreasing=true)
	sets ind so that vec(ind(0))<=vec(ind(1))<=...<=vec(ind(vec.dim()-1)) (vec may be rectangular, set nondecreasing=false for opposite order)
std::ostream &	operator<< (std::ostream &o, const Matrix &v)
	output format (nr and nc are Integer values, all others Real values): nr nc \n A(1,1) A(1,2) ... A(1,nc) \n A(2,1) ... A(nr,nc) \n
std::istream &	operator>> (std::istream &i, Matrix &v)
	input format (nr and nc are Integer values, all others Real values): nr nc \n A(1,1) A(1,2) ... A(1,nc) \n A(2,1) ... A(nr,nc) \n
Real	ip (const Matrix &A, const Matrix &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Real	colip (const Matrix &A, Integer j, const Matrix *scaling=0)
	returns the squared Frobenius norm of column j of A, i.e., the sum of A(i,j)*A(i,j) over all i with possibly (if scaling!=0) each term i multiplied by (*scaling)(i)
Real	rowip (const Matrix &A, Integer i, const Matrix *scaling=0)
	returns the squared Frobenius norm of row i of A, i.e., the sum of A(i,j)*A(i,j) over all j with possibly (if scaling!=0) each term j multiplied by (*scaling)(j)
Matrix	colsip (const Matrix &A)
	returns the column vector of the squared Frobenius norm of all columns j of A, i.e., the sum of A(i,j)*A(i,j) over all i for each j
Matrix	rowsip (const Matrix &A)
	returns the row vector of the squared Frobenius norm of all rowd i of A, i.e., the sum of A(i,j)*A(i,j) over all j for each i
Real	norm2 (const Matrix &A)
	returns the Frobenius norm of A, i.e., the square root of the sum of A(i,j)*A(i,j) over all i,j
Matrix	operator* (const Matrix &A, const Matrix &B)
	returns Matrix equal to A*B
Matrix	operator+ (const Matrix &A, const Matrix &B)
	returns Matrix equal to A+B
Matrix	operator- (const Matrix &A, const Matrix &B)
	returns Matrix equal to A-B
Matrix	operator% (const Matrix &A, const Matrix &B)
	ATTENTION: this is redefined as the Hadamard product, C(i,j)=A(i,j)*B(i,j) for all i,j.
Matrix	operator/ (const Matrix &A, const Matrix &B)
	ATTENTION: this is redefined to act componentwise without checking for zeros, C(i,j)=A(i,j)/B(i,j) for all i,j.
Matrix	operator* (const Matrix &A, Real d)
	returns Matrix equal to A*d
Matrix	operator* (Real d, const Matrix &A)
	returns Matrix equal to d*A
Matrix	operator/ (const Matrix &A, Real d)
	returns Matrix equal to A/d; ATTENTION: d is NOT checked for 0 More...
Matrix	operator+ (const Matrix &A, Real d)
	returns (i,j)=A(i,j)+d for all i,j
Matrix	operator+ (Real d, const Matrix &A)
	returns (i,j)=A(i,j)+d for all i,j
Matrix	operator- (const Matrix &A, Real d)
	returns (i,j)=A(i,j)-d for all i,j
Matrix	operator- (Real d, const Matrix &A)
	returns (i,j)=d-A(i,j) for all i,j
int	QR_factor (const Matrix &A, Matrix &Q, Matrix &R, Real tol)
	computes a Householder QR factorization of A and outputs Q and R leaving A unchanged; always returns 0
int	QR_factor (const Matrix &A, Matrix &Q, Matrix &R, Indexmatrix &piv, Real tol)
	computes a Householder QR factorization of A with pivating. It outputs Q, R, and the pivoting permuation in piv; returns the rank of A
Matrix	triu (const Matrix &A, Integer i=0)
	retuns a matrix that keeps the upper triangle of A starting with diagonal d, i.e., (i,j)=A(i,j) for 0<=i<row dimension, max(0,i+d)<=j<column dimension, and sets (i,j)=0 otherwise
Matrix	tril (const Matrix &A, Integer i=0)
	retuns a matrix that keeps the lower triangle of A starting with diagonal d, i.e., (i,j)=A(i,j) for 0<=i<row dimension, 0<=j<min(i+d+1,column dimension), and sets (i,j)=0 otherwise
Matrix	concat_right (const Matrix &A, const Matrix &B)
	returns a new matrix [A, B], i.e., it concats matrices A and B rowwise; A or B may be a 0x0 matrix
Matrix	concat_below (const Matrix &A, const Matrix &B)
	returns a bew matrix [A; B], i.e., it concats matrices A and B columnwise; A or B may be a 0x0 matrix
void	swap (Matrix &A, Matrix &B)
	swap the content of the two matrices A and B (involves no copying)
Matrix	rand (Integer rows, Integer cols, CH_Tools::GB_rand *random_generator=0)
	return a nr x nc matrix with (i,j) assigned a random number uniformly from [0,1] for all i,j
Matrix	inv (const Matrix &A)
	returns a matrix with elements (i,j)=1./((*this)(i,j)) for all i,j; ATTENTION: no check for division by zero
Matrix	sqrt (const Matrix &A)
	returns a matrix with elements (i,j)=sqrt((*this)(i,j)) for all i,j
Matrix	sqr (const Matrix &A)
	returns a matrix with elements (i,j)=sqr((*this)(i,j)) for all i,j
Matrix	sign (const Matrix &A, Real tol=1e-12)
	returns a matrix with elements (i,j)=sign((*this)(i,j)) for all i,j using sign(double,double)
Matrix	floor (const Matrix &A)
	returns a matrix with elements (i,j)=floor((*this)(i,j)) for all i,j
Matrix	ceil (const Matrix &A)
	returns a matrix with elements (i,j)=ceil((*this)(i,j)) for all i,j
Matrix	rint (const Matrix &A)
	returns a matrix with elements (i,j)=rint((*this)(i,j)) for all i,j
Matrix	round (const Matrix &A)
	returns a matrix with elements (i,j)=round((*this)(i,j)) for all i,j
Matrix	operator> (const Matrix &A, const Matrix &B)
	returns a matrix having elements (i,j)=Real(A(i,j)>d) for all i,j More...
Matrix	operator>= (const Matrix &A, const Matrix &B)
	returns a matrix having elements (i,j)=Real(A(i,j)>=d) for all i,j More...
Matrix	operator< (Real d, const Matrix &A)
	returns a matrix having elements (i,j)=Real(d<A(i,j)) for all i,j
Matrix	operator> (Real d, const Matrix &A)
	returns a matrix having elements (i,j)=Real(d>A(i,j)) for all i,j
Matrix	operator<= (Real d, const Matrix &A)
	returns a matrix having elements (i,j)=Real(d<=A(i,j)) for all i,j
Matrix	operator>= (Real d, const Matrix &A)
	returns a matrix having elements (i,j)=Real(d>=A(i,j)) for all i,j
Matrix	operator== (Real d, const Matrix &A)
	returns a matrix having elements (i,j)=Real(d==A(i,j)) for all i,j
Matrix	operator!= (Real d, const Matrix &A)
	returns a matrix having elements (i,j)=Real(d!=A(i,j)) for all i,j
bool	equal (const Matrix &A, const Matrix &B)
	returns true if both matrices have the same size and the same elements
Indexmatrix	find (const Matrix &A, Real tol=1e-10)
	returns an Indexmatrix ind so that A(ind(i)) 0<=i<ind.dim() runs through all nonzero elements with abs(A(j))>tol
Indexmatrix	find_number (const Matrix &A, Real num=0., Real tol=1e-10)
	returns an Indexmatrix ind so that A(ind(i)) 0<=i<ind.dim() runs through all elements of A having value num, i.e., abs(A(j)-num)<tol
Matrix	diag (const Symmatrix &A)
	returns a column vector v consisting of the elements v(i)=(*this)(i,i), 0<=i<row dimension
Symmatrix	Diag (const Matrix &A)
	returns a symmetric diagonal matrix S of order A.dim() with vec(A) on the diagonal, i.e., S(i,i)=A(i) for all i and S(i,j)=0 for i!=j
Symmatrix &	rankadd (const Matrix &A, Symmatrix &C, Real alpha=1., Real beta=0., int trans=0)
	returns C=beta*C+alpha* A*A^T, where A may be transposed. If beta==0. then C is initiliazed to the correct size.
Symmatrix &	scaledrankadd (const Matrix &A, const Matrix &D, Symmatrix &C, Real alpha=1., Real beta=0., int trans=0)
	returns C=beta*C+alpha* A*D*A^T, where D is a vector representing a diagonal matrix and A may be transposed; if beta==0. then C is initialized to the correct size
Symmatrix &	rank2add (const Matrix &A, const Matrix &B, Symmatrix &C, Real alpha=1., Real beta=0., int trans=0)
	returns C=beta*C+alpha*(A*B^T+B*A^T)/2 [or for transposed (A^T*B+B^T*A)/2]. If beta==0. then C is initiliazed to the correct size.
Matrix &	genmult (const Symmatrix &A, const Sparsemat &B, Matrix &C, Real alpha=1., Real beta=0., int btrans=0)
	returns C=beta*C+alpha*A*B, where B may be transposed; if beta==0. then C is initialized to the correct size
Matrix &	genmult (const Sparsemat &A, const Symmatrix &B, Matrix &C, Real alpha=1., Real beta=0., int atrans=0)
	returns C=beta*C+alpha*A*B, where A may be transposed; if beta==0. then C is initialized to the correct size
Symmatrix &	rankadd (const Sparsemat &A, Symmatrix &C, Real alpha=1., Real beta=0., int trans=0)
	returns C=beta*C+alpha* A*A^T, where A may be transposed; if beta==0. then C is initialized to the correct size
Symmatrix &	scaledrankadd (const Sparsemat &A, const Matrix &D, Symmatrix &C, Real alpha=1., Real beta=0., int trans=0)
	returns C=beta*C+alpha* A*D*A^T, where D is a vector representing a diagonal matrix and A may be transposed; if beta==0. then C is initialized to the correct size
Symmatrix &	rank2add (const Sparsemat &A, const Matrix &B, Symmatrix &C, Real alpha=1., Real beta=0., int trans=0)
	returns C=beta*C+alpha*(A*B^T+B*A^T)/2 [or for transposed (A^T*B+B^T*A)/2]. If beta==0. then C is initiliazed to the correct size.
Symmatrix	abs (const Symmatrix &A)
	returns a Symmatrix with elements abs(A(i,j))
Real	trace (const Symmatrix &A)
	returns the sum of the diagonal elements A(i,i) over all i
Real	ip (const Symmatrix &A, const Symmatrix &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Real	ip (const Matrix &A, const Symmatrix &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Real	ip (const Symmatrix &A, const Matrix &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Matrix	sumrows (const Symmatrix &A)
	returns a row vector holding the sum over all rows, i.e., (1 1 ... 1)*A
Matrix	sumcols (const Symmatrix &A)
	returns a column vector holding the sum over all columns, i.e., A*(1 1 ... 1)^T
Real	sum (const Symmatrix &A)
	returns the sum over all elements of A, i.e., (1 1 ... 1)*A*(1 1 ... 1)^T
void	svec (const Symmatrix &A, Matrix &sv, Real a=1., bool add=false, Integer startindex_vec=-1, Integer startindex_A=0, Integer blockdim=-1)
	the symmetric vec operator stacks the lower triangle of A to a n*(n+1)/2 vector with the same norm2 as A; here it sets svec(A)=[a11,sqrt(2)a12,...,sqrt(2)a1n,a22,...,sqrt(2)a(n-1,n),ann]', multiplies it by a and sets or adds (if add==true) it to v starting from startindex_vec possibly restricted to the subblock of order blockdim (whenever >=0, else blockdim is set to A.rowdim()-startindex_A) starting from startindex_A (must be >=0); if add==false and startindex_vec<0 then vec is also reinitialzed to the appropriate size
void	sveci (const Matrix &sv, Symmatrix &A, Real a=1., bool add=false, Integer startindex_vec=0, Integer startindex_A=-1, Integer blockdim=-1)
	the inverse operator to svec, extracts from v at startindex_vec (>=0) the symmetric matrix of blockdim adding its mutliple by a into A starting at startindex_A; if add==false and startindex_A<0 A is initialized to the size of blockdim; if the latter is also negative then v.dim()-startindex_vec must match an exact order and matrix A is initialized to this size. In all other cases the size of the symmetric matrix determines the missing parameters and vec.dim-startindex_vec
void	skron (const Symmatrix &A, const Symmatrix &B, Symmatrix &S, Real alpha=1., bool add=false, Integer startindex_S=-1)
	def symmetric Kronecker product (A skron B)svec(C)=(BCA'+ACB')/2; sets S=alpha*(A skron B) or S*=... (if add==true) possibly shifted to the block starting at startindex_S; if add==false and startindex_S<0, S is initialzed to the correct size
void	symscale (const Symmatrix &A, const Matrix &B, Symmatrix &S, Real alpha=1., Real beta=0., int btrans=0)
	sets S=beta*S+alpha*B'*A*B for symmatrix A and matrix B
Matrix	minrows (const Symmatrix &A)
	returns a row vector holding in each column the minimum over all rows in this column
Matrix	mincols (const Symmatrix &A)
	returns a column vector holding in each row the minimum over all columns in this row
Real	min (const Symmatrix &A)
	returns the minimum value over all elements of the matrix
Matrix	maxrows (const Symmatrix &A)
	returns a row vector holding in each column the maximum over all rows in this column
Matrix	maxcols (const Symmatrix &A)
	returns a column vector holding in each row the maximum over all columns in this row
Real	max (const Symmatrix &A)
	returns the maximum value over all elements of the matrix
std::ostream &	operator<< (std::ostream &o, const Symmatrix &A)
	output format (nr and nc are Integer values, all others Real values): nr nc \n A(1,1) A(1,2) ... A(1,nc) \n A(2,1) ... A(nr,nc) \n
std::istream &	operator>> (std::istream &i, Symmatrix &A)
	input format (nr and nc are Integer values, all others Real values): nr nc \n A(1,1) A(1,2) ... A(1,nc) \n A(2,1) ... A(nr,nc) \n
void	swap (Symmatrix &A, Symmatrix &B)
	swap the content of the two matrices A and B (involves no copying)
Symmatrix &	xbpeya (Symmatrix &x, const Symmatrix &y, Real alpha=1., Real beta=0.)
	returns x= alpha*y+beta*x; if beta==0. then x is initialized to the correct size
Symmatrix &	xeyapzb (Symmatrix &x, const Symmatrix &y, const Symmatrix &z, Real alpha=1., Real beta=1.)
	returns x= alpha*y+beta*z; x is initialized to the correct size
Matrix	operator* (const Symmatrix &A, const Symmatrix &B)
	returns a Matrix that equals A*B
Symmatrix	operator% (const Symmatrix &A, const Symmatrix &B)
	returns a Matrix that equals AB (where % is overloaded as elementwise multiplication) More...
Symmatrix	operator+ (const Symmatrix &A, const Symmatrix &B)
	returns a Matrix that equals A+B
Symmatrix	operator- (const Symmatrix &A, const Symmatrix &B)
	returns a Matrix that equals A-B
Matrix	operator* (const Symmatrix &A, const Matrix &B)
	returns a Matrix that equals A*B
Matrix	operator* (const Matrix &A, const Symmatrix &B)
	returns a Matrix that equals A*B
Matrix	operator+ (const Symmatrix &A, const Matrix &B)
	returns a Matrix that equals A+B
Matrix	operator+ (const Matrix &A, const Symmatrix &B)
	returns a Matrix that equals A+B
Matrix	operator- (const Symmatrix &A, const Matrix &B)
	returns a Matrix that equals A-B
Matrix	operator- (const Matrix &A, const Symmatrix &B)
	returns a Matrix that equals A-B
Symmatrix	operator* (const Symmatrix &A, Real d)
	returns a Symmatrix that equals A*d
Symmatrix	operator* (Real d, const Symmatrix &A)
	returns a Symmatrix that equals A*d
Symmatrix	operator/ (const Symmatrix &A, Real d)
	returns a Symmatrix that equals A/d; ATTENTION: no check against division by zero
Symmatrix	operator+ (const Symmatrix &A, Real d)
	returns a Symmatrix that equals A+d (d is added to each element) More...
Symmatrix	operator+ (Real d, const Symmatrix &A)
	returns a Symmatrix that equals A+d (d is added to each element) More...
Symmatrix	operator- (const Symmatrix &A, Real d)
	returns a Symmatrix that equals A-d (d is subtracted from each element) More...
Symmatrix	operator- (Real d, const Symmatrix &A)
	returns a Symmatrix that equals d-A (each element subtracted from d) More...
Matrix	svec (const Symmatrix &A)
	the symmetric vec operator, stacks the lower triangle of A to a n*(n+1)/2 vector with the same norm2 as A; i.e., it returns svec(A)=[a11,sqrt(2)a12,...,sqrt(2)a1n,a22,...,sqrt(2)a(n-1,n),ann]'
Symmatrix	skron (const Symmatrix &A, const Symmatrix &B, Real alpha=1., bool add=false, Integer startindex_S=-1)
	the symmetric Kronecker product, defined via (A skron B)svec(C)=(BCA'+ACB')/2; sets or adds (if add==true) the symmetric matrix a*(A skron B) into S starting at startindex_S; if add==false and startindex_S<0, S is initialzed to the correct size
Real	norm2 (const Symmatrix &A)
	returns the Frobenius norm of A, i.e., the square root of the sum of A(i,j)*A(i,j) over all i,j
Symmatrix	transpose (const Symmatrix &A)
	returns a copy of A (drop it or use a constructor instead) More...
void	swap (Sparsemat &A, Sparsemat &B)
	swap the content of the two sparse matrices A and B (involves no copying)
Sparsemat &	xbpeya (Sparsemat &x, const Sparsemat &y, Real alpha=1., Real beta=0., int ytrans=0)
	returns x= alpha*y+beta*x, where y may be transposed (ytrans=1); if beta==0. then x is initialized to the correct size
Matrix &	genmult (const Sparsemat &A, const Sparsemat &B, Matrix &C, Real alpha=1., Real beta=0., int atrans=0, int btrans=0)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size
Sparsemat	operator* (const Sparsemat &A, const Sparsemat &B)
	returns a Sparsemat equal to A*B
Matrix &	genmult (const Sparsesym &A, const Sparsemat &B, Matrix &C, Real alpha=1., Real beta=0., int btrans=0)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size More...
Matrix &	genmult (const Sparsemat &A, const Sparsesym &B, Matrix &C, Real alpha=1., Real beta=0., int atrans=0)
	returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size More...
Sparsemat	abs (const Sparsemat &A)
	returns a Sparsmat with elements abs((*this)(i,j)) for all i,j
Real	trace (const Sparsemat &A)
	returns the sum of the diagonal elements A(i,i) over all i
Real	ip (const Sparsemat &A, const Sparsemat &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Real	ip (const Sparsemat &A, const Matrix &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Real	colip (const Sparsemat &A, Integer j, const Matrix *scaling=0)
	returns the squared Frobenius norm of col i of A, i.e., the sum of A(i,j)*A(i,j) over all i with possibly (if scaling!=0) each term i multiplied by (*scaling)(i)
Real	rowip (const Sparsemat &A, Integer i, const Matrix *scaling=0)
	returns the squared Frobenius norm of row i of A, i.e., the sum of A(i,j)*A(i,j) over all j with possibly (if scaling!=0) each term j multiplied by (*scaling)(j)
Sparsemat	colsip (const Sparsemat &A, const Matrix *colscaling=0)
	returns the row vector of the squared Frobenius norm of all columns j of A, i.e., the sum of A(i,j)*A(i,j) over all i for each j with possibly (if scaling!=0) each term i multiplied by (*scaling)(i) More...
Sparsemat	rowsip (const Sparsemat &A, const Matrix *rowscaling=0)
	returns the column vector of the squared Frobenius norm of all rows i of A, i.e., the sum of A(i,j)*A(i,j) over all j for each i with possibly (if scaling!=0) each term j multiplied by (*scaling)(j) More...
Matrix	sumrows (const Sparsemat &A)
	returns a row vector holding the sum over all rows, i.e., (1 1 ... 1)*A
Matrix	sumcols (const Sparsemat &A)
	returns a column vector holding the sum over all columns, i.e., A*(1 1 ... 1)^T
std::ostream &	operator<< (std::ostream &o, const Sparsemat &v)
	output format: nr nc nz \n i1 j1 val1\n i2 j2 val2\n ... inz jnz valnz\n
std::istream &	operator>> (std::istream &i, Sparsemat &v)
	input format: nr nc nz \n i1 j1 val1\n i2 j2 val2\n ... inz jnz valnz\n
Sparsemat	concat_right (const Sparsemat &A, const Sparsemat &B)
	returns a new sparse matrix [A, B], i.e., it concats matrices A and B rowwise; A or B may be a 0x0 matrix
Sparsemat	concat_below (const Sparsemat &A, const Sparsemat &B)
	returns a new sparse matrix [A; B], i.e., it concats matrices A and B columnwise; A or B may be a 0x0 matrix
Real	norm2 (const Sparsemat &A)
	returns the Frobenius norm of A, i.e., the square root of the sum of A(i,j)*A(i,j) over all i,j
Sparsemat	operator+ (const Sparsemat &A, const Sparsemat &B)
	returns a Sparsemat equal to A+B
Sparsemat	operator- (const Sparsemat &A, const Sparsemat &B)
	returns a Sparsemat equal to A-B
Sparsemat	operator* (const Sparsemat &A, Real d)
	returns a Sparsemat equal to A*d
Sparsemat	operator* (Real d, const Sparsemat &A)
	returns a Sparsemat equal to d*A
Sparsemat	operator/ (const Sparsemat &A, Real d)
	returns a Sparsemat equal to A/d; ATTENTION: no check against division by zero More...
Matrix	operator* (const Sparsemat &A, const Matrix &B)
	returns a Matrix equal to A*B
Matrix	operator* (const Matrix &A, const Sparsemat &B)
	returns a Matrix equal to A*B
Matrix	operator+ (const Sparsemat &A, const Matrix &B)
	returns a Matrix equal to A+B
Matrix	operator+ (const Matrix &A, const Sparsemat &B)
	returns a Matrix equal to A+B
Matrix	operator- (const Sparsemat &A, const Matrix &B)
	returns a Matrix equal to A-B
Matrix	operator- (const Matrix &A, const Sparsemat &B)
	returns a Matrix equal to A-B
Real	ip (const Matrix &A, const Sparsemat &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Real	sum (const Sparsemat &A)
	returns the sum over all elements of A, i.e., (1 1 ... 1)*A*(1 1 ... 1)^T
Sparsemat	transpose (const Sparsemat &A)
	returns a Sparsemat that is the transpose of A
Matrix	diag (const Sparsesym &A)
	returns the diagonal of A as a dense Matrix vector
Sparsesym	sparseDiag (const Matrix &A, Real tol=SPARSE_ZERO_TOL)
	forms a sparse symmetrix matrix having vector A on its diagonal
void	swap (Sparsesym &A, Sparsesym &B)
	swap the content of the two sparse matrices A and B (involves no copying)
Sparsesym &	xeyapzb (Sparsesym &x, const Sparsesym &y, const Sparsesym &z, Real alpha=1., Real beta=1.)
	returns x= alpha*y+beta*z; x is initialized to the correct size
Sparsesym &	support_rankadd (const Matrix &A, Sparsesym &C, Real alpha=1., Real beta=0., int trans=0)
	returns C=beta*C+alpha*AA^T (or A^TA), but only on the current support of C
Real	ip (const Symmatrix &A, const Sparsesym &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Sparsesym	abs (const Sparsesym &A)
	returns a Sparsesym with elements abs((*this)(i,j)) for all i,j
Real	trace (const Sparsesym &A)
	returns the sum of the diagonal elements A(i,i) over all i
Real	ip (const Sparsesym &A, const Sparsesym &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Real	ip (const Matrix &A, const Sparsesym &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Real	norm2 (const Sparsesym &A)
	returns the Frobenius norm of A, i.e., the square root of the sum of A(i,j)*A(i,j) over all i,j
Matrix	sumrows (const Sparsesym &A)
	returns a row vector holding the sum over all rows, i.e., (1 1 ... 1)*A
Real	sum (const Sparsesym &A)
	returns the sum over all elements of A, i.e., (1 1 ... 1)*A*(1 1 ... 1)^T
std::ostream &	operator<< (std::ostream &o, const Sparsesym &v)
	output format (lower triangle): nr nz \n i1 j1 val1\n i2 j2 val2\n ... inz jnz valnz\n
std::istream &	operator>> (std::istream &i, Sparsesym &v)
	input format (lower triangle): nr nz \n i1 j1 val1\n i2 j2 val2\n ... inz jnz valnz\n
Sparsesym &	xbpeya (Sparsesym &x, const Sparsesym &y, Real alpha=1., Real beta=0.)
	returns x= alpha*y+beta*x; if beta==0. then x is initialized to the correct size
Sparsesym	operator+ (const Sparsesym &A, const Sparsesym &B)
	returns a Sparsesym that equals A+B
Sparsesym	operator- (const Sparsesym &A, const Sparsesym &B)
	returns a Sparsesym that equals A-B
Sparsesym	operator* (const Sparsesym &A, Real d)
	returns a Sparsesym that equals A*d
Sparsesym	operator* (Real d, const Sparsesym &A)
	returns a Sparsesym that equals A*d
Sparsesym	operator/ (const Sparsesym &A, Real d)
	returns a Sparsesym that equals A/d; ATTENTION: no check for devision by zero More...
Matrix	operator* (const Sparsesym &A, const Matrix &B)
	returns a Matrix that equals A*B
Matrix	operator* (const Matrix &A, const Sparsesym &B)
	returns a Matrix that equals A*B
Matrix	operator+ (const Matrix &A, const Sparsesym &B)
	returns a Matrix that equals A+B
Matrix	operator+ (const Sparsesym &A, const Matrix &B)
	returns a Matrix that equals A+B
Matrix	operator- (const Matrix &A, const Sparsesym &B)
	returns a Matrix that equals A-B
Matrix	operator- (const Sparsesym &A, const Matrix &B)
	returns a Matrix that equals A-B
Real	ip (const Sparsesym &A, const Matrix &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Sparsesym	transpose (const Sparsesym &A)
	returns a copy of A (drop it or use a constructor instead) More...
Matrix	sumcols (const Sparsesym &A)
	returns a column vector holding the sum over all columns, i.e., A*(1 1 ... 1)^T
Symmatrix	operator+ (const Sparsesym &A, const Symmatrix &B)
	returns a Symmatrix that equals A+B
Symmatrix	operator+ (const Symmatrix &A, const Sparsesym &B)
	returns a Symmatrix that equals A+B
Symmatrix	operator- (const Sparsesym &A, const Symmatrix &B)
	returns a Symmatrix that equals A-B
Symmatrix	operator- (const Symmatrix &A, const Sparsesym &B)
	returns a Symmatrix that equals A-B
Real	ip (const Sparsesym &A, const Symmatrix &B)
	returns the usual inner product of A and B, i.e., the sum of A(i,j)*B(i,j) over all i,j
Matrix	operator* (const Sparsesym &A, const Sparsemat &B)
	returns a Matrix that equals A*B
Matrix	operator* (const Sparsemat &A, const Sparsesym &B)
	returns a Matrix that equals A*B
Matrix	operator* (const Symmatrix &A, const Sparsemat &B)
	returns a Matrix that equals A*B
Matrix	operator* (const Sparsemat &A, const Symmatrix &B)
	returns a Matrix that equals A*B
Equal (Members)
int	equal (const Sparsemat &A, const Sparsemat &B, Real eqtol=1e-10)
	returns 1 if both matrices are identical, 0 otherwise
int	equal (const Sparsesym &A, const Sparsesym &B, Real eqtol=1e-10)
	returns 1 if both matrices are identical, 0 otherwise

Variables
const Integer	max_Integer = INT_MAX
	maximal attainable value by an Integer
const Integer	min_Integer = INT_MIN
	minimal attainable value by an Integer
const Real	max_Real = DBL_MAX
	maximal attainable value by a Real
const Real	min_Real = -DBL_MAX
	minimal attainable value by a Real
const Real	eps_Real = DBL_EPSILON
	machine epsilon for type Real
CH_Tools::GB_rand	mat_randgen
	common random number generator for use when a random matrix is generated. More...
std::default_random_engine	mat_std_randgen
	optional fast alternative random number generator from std
std::mt19937	mat_mt_randgen
	optional high quality alternative random number generator from std
std::mt19937_64	mat_mt64_randgen
	optional high quality alternative random number generator from std
std::ostream *	materrout
	if not zero, this is the output stream for runtime error messages, by default it is set to &std::cout

Detailed Description

Matrix Classes and Linear Algebra. See Matrix Classes (namespace CH_Matrix_Classes) for a quick introduction.

Function Documentation

colsip()

Sparsemat CH_Matrix_Classes::colsip	(	const Sparsemat &	A,
		const Matrix *	colscaling = 0
	)

returns the row vector of the squared Frobenius norm of all columns j of A, i.e., the sum of A(i,j)*A(i,j) over all i for each j with possibly (if scaling!=0) each term i multiplied by (*scaling)(i)

returns the column vector of the squared Frobenius norm of all columns j of A, i.e., the sum of A(i,j)*A(i,j) over all i for each j with possibly (if scaling~=0) each term i multiplied by (*scaling)(i)

genmult() [1/6]

Matrix& CH_Matrix_Classes::genmult	(	const Sparsesym &	A,
		const Sparsemat &	B,
		Matrix &	C,
		Real	alpha = 1.,
		Real	beta = 0.,
		int	btrans = 0
	)

returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size

returns C=beta*C+alpha*A*B, where B may be transposed; if beta==0. then C is initialized to the correct size

genmult() [2/6]

Matrix& CH_Matrix_Classes::genmult	(	const Sparsemat &	A,
		const Sparsesym &	B,
		Matrix &	C,
		Real	alpha = 1.,
		Real	beta = 0.,
		int	atrans = 0
	)

returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size

returns C=beta*C+alpha*A*B, where A may be transposed; if beta==0. then C is initialized to the correct size

genmult() [3/6]

Matrix & CH_Matrix_Classes::genmult	(	const Symmatrix &	A,
		const Matrix &	B,
		Matrix &	C,
		Real	alpha,
		Real	beta,
		int	btrans
	)

returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size

returns C=beta*C+alpha*A*B, where B may be transposed; C must not be equal to B; if beta==0. then C is initialized to the correct size

genmult() [4/6]

Matrix & CH_Matrix_Classes::genmult	(	const Matrix &	A,
		const Symmatrix &	B,
		Matrix &	C,
		Real	alpha,
		Real	beta,
		int	atrans
	)

returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size

returns C=beta*C+alpha*A*B, where A may be transposed; C must not be equal to A; if beta==0. then C is initialized to the correct size

genmult() [5/6]

Matrix& CH_Matrix_Classes::genmult	(	const Sparsesym &	A,
		const Matrix &	B,
		Matrix &	C,
		Real	alpha,
		Real	beta,
		int	btrans
	)

returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size

returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to B; if beta==0. then C is initialized to the correct size

genmult() [6/6]

Matrix& CH_Matrix_Classes::genmult	(	const Matrix &	A,
		const Sparsesym &	B,
		Matrix &	C,
		Real	alpha,
		Real	beta,
		int	atrans
	)

returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to A and B; if beta==0. then C is initialized to the correct size

returns C=beta*C+alpha*A*B, where A and B may be transposed; C must not be equal to B; if beta==0. then C is initialized to the correct size

operator%()

Symmatrix CH_Matrix_Classes::operator% ( const Symmatrix &  A, const Symmatrix &  B  )

inline

returns a Matrix that equals AB (where % is overloaded as elementwise multiplication)

ATTENTION: this is redefined as the Hadamard product and sets (i,j)=A(i,j)*B(i,j) for all i<=j.

operator+() [1/2]

Symmatrix CH_Matrix_Classes::operator+ ( const Symmatrix &  A, Real  d  )

inline

returns a Symmatrix that equals A+d (d is added to each element)

returns (i,j)=A(i,j)+d for all i<=j

operator+() [2/2]

Symmatrix CH_Matrix_Classes::operator+ ( Real  d, const Symmatrix &  A  )

inline

returns a Symmatrix that equals A+d (d is added to each element)

returns (i,j)=A(i,j)+d for all i<=j

operator-() [1/2]

Symmatrix CH_Matrix_Classes::operator- ( const Symmatrix &  A, Real  d  )

inline

returns a Symmatrix that equals A-d (d is subtracted from each element)

returns (i,j)=A(i,j)-d for all i<=j

operator-() [2/2]

Symmatrix CH_Matrix_Classes::operator- ( Real  d, const Symmatrix &  A  )

inline

returns a Symmatrix that equals d-A (each element subtracted from d)

returns (i,j)=d-A(i,j) for all i<=j

Referenced by CH_Matrix_Classes::Symmatrix::get_store(), CH_Matrix_Classes::Symmatrix::operator%=(), and CH_Matrix_Classes::Symmatrix::transpose().

operator/() [1/3]

Sparsesym CH_Matrix_Classes::operator/ ( const Sparsesym &  A, Real  d  )

inline

returns a Sparsesym that equals A/d; ATTENTION: no check for devision by zero

ATTENTION: d is NOT checked for 0.

References CH_Matrix_Classes::Sparsesym::Sparsesym().

operator/() [2/3]

Sparsemat CH_Matrix_Classes::operator/ ( const Sparsemat &  A, Real  d  )

inline

returns a Sparsemat equal to A/d; ATTENTION: no check against division by zero

ATTENTION: d is NOT checked for 0.

References CH_Matrix_Classes::Sparsemat::Sparsemat().

operator/() [3/3]

Matrix CH_Matrix_Classes::operator/ ( const Matrix &  A, Real  d  )

inline

returns Matrix equal to A/d; ATTENTION: d is NOT checked for 0

ATTENTION: d is NOT checked for 0.

References CH_Matrix_Classes::Matrix::Matrix().

operator>()

Matrix CH_Matrix_Classes::operator> ( const Matrix &  A, const Matrix &  B  )

inline

returns a matrix having elements (i,j)=Real(A(i,j)>d) for all i,j

returns a matrix having elements (i,j)=Real(A(i,j)>B(i,j)) for all i,j

operator>=()

Matrix CH_Matrix_Classes::operator>= ( const Matrix &  A, const Matrix &  B  )

inline

returns a matrix having elements (i,j)=Real(A(i,j)>=d) for all i,j

returns a matrix having elements (i,j)=Real(A(i,j)>=B(i,j)) for all i,j

rowsip()

Sparsemat CH_Matrix_Classes::rowsip	(	const Sparsemat &	A,
		const Matrix *	rowscaling = 0
	)

returns the column vector of the squared Frobenius norm of all rows i of A, i.e., the sum of A(i,j)*A(i,j) over all j for each i with possibly (if scaling!=0) each term j multiplied by (*scaling)(j)

returns the row vector of the squared Frobenius norm of all rows i of A, i.e., the sum of A(i,j)*A(i,j) over all j for each i with possibly (if scaling~=0) each term j multiplied by (*scaling)(j)

trace()

Real CH_Matrix_Classes::trace

(

const Matrix &

A

)

=sum(diag(A))

returns the sum of the diagonal elements A(i,i) over all i

transpose() [1/2]

Sparsesym CH_Matrix_Classes::transpose ( const Sparsesym &  A )

inline

returns a copy of A (drop it or use a constructor instead)

(drop it or use a constructor instead)

References CH_Matrix_Classes::Sparsesym::Sparsesym().

transpose() [2/2]

Symmatrix CH_Matrix_Classes::transpose ( const Symmatrix &  A )

inline

returns a copy of A (drop it or use a constructor instead)

(drop it or use a constructor instead)

References CH_Matrix_Classes::Symmatrix::Symmatrix().