#include <rw/chbndmat.h> //DComplexHermBandMat DComplexHermBandMat H;
The class DComplexHermBandMat encapsulates a Hermitian banded matrix. A Hermitian banded matrix is Hermitian, and nonzero only near the diagonal. Specifically, if h is the half bandwidth, then any entries for which i-j>h or j-i>h are defined to be 0; in addition, an entry Aij = conj(Aji). Although this strict definition implies a matrix with 0 imaginary parts along the diagonal, the Rogue Wave class implementation allows arbitrary diagonal entries.
#include <rw/chbndmat.h>
main()
{
DComplexHermBandMat HB(5,5,1);
HB.zero();
HB.set(2, 2, DComplex(2,1));
}
As an example of the storage scheme for banded Hermitian matrices, consider the following 9x9 matrix with a half bandwidth of 2, where entries aij are notation for conj(Aij):
The upper triangle is stored column by column. For convenience, there are some unused locations in the vector of data. These are indicated as XXX in the following illustration of the storage vector:
[ XXX XXX A11 XXX A12 A22 A13 A23 A33 A24 A34 A44 A35 A45
A55 A46 A56 A66 A57 A67 A77 A68 A78 A88 A79 A89 A99 ]
The mapping between the array and the storage vector is as follows:
DComplexHermBandMat();
Default constructor. Builds a matrix of size 0 x 0. This constructor is necessary to declare a matrix with no explicit constructor or an array of matrices.
DComplexHermBandMat(const DComplexHermBandMat& A);
Builds a copy of its argument, A. Note that the new matrix references A's data. To construct a matrix with its own copy of the data, use either the copy or deepenShallowCopy member functions.
DComplexHermBandMat(unsigned n, unsigned n, unsigned hb);
Defines an uninitialized matrix of size n x n with half bandwidth hb.
DComplexHermBandMat(const RWMathVec<DComplex>& vd, unsigned n,
unsigned n, unsigned hb);
Constructs a size n x n matrix with half bandwidth hb using the data in the passed vector. This data must be stored in the format described in the Storage Scheme section. The resultant matrix references the data in
vector vd.
DComplexHermBandMat(const DoubleSymBandMat& re);
Converts a real matrix to a Hermitian banded matrix.
unsigned DComplexHermBandMat::bandwidth();
Returns the bandwidth of the matrix.
ROCJDComplexRef DComplexHermBandMat::bcref(int i, int j);
Returns a reference to the ijth element of the matrix, after doing bounds checking.
void DComplexHermBandMat::bcset(int i, int j, DComplex x);
Sets the ijth element of the matrix equal to x, after doing bounds checking.
DComplex DComplexHermBandMat::bcval(int i, int j);
Returns the value of the ijth element of the matrix, after doing bounds checking.
unsigned DComplexHermBandMat::binaryStoreSize();
Returns the number of bytes that it would take to write the matrix to a file using saveOn.
unsigned DComplexHermBandMat::cols();
Returns the number of columns in the matrix.
DComplexHermBandMat DComplexHermBandMat::copy();
Creates a copy of this matrix with distinct data. The stride of the data vector in the new matrix is guaranteed to be 1.
DComplex* DComplexHermBandMat::data();
Returns a pointer to the first item of data in the vector storing the matrix's data. You can use this to pass the matrix's data to C or FORTRAN subroutines, but be aware that the stride of the data vector may not be 1.
RWMathVec<DComplex> DComplexHermBandMat::dataVec();
Returns the matrix's data vector. This is where the explicitly stored entries in the matrix are kept.
DComplexHermBandMat DComplexHermBandMat::deepCopy();
Creates a copy of this matrix with distinct data. The stride of the data vector in the new matrix is guaranteed to be 1.
void DComplexHermBandMat::deepenShallowCopy();
Ensures that the data in the matrix is not shared by any other matrix or vector. Also ensures that the stride in the data vector is equal to 1. If necessary, a new copy of the data vector is made.
unsigned DComplexHermBandMat::halfBandwidth();
Returns the half bandwidth of the matrix.
DComplexHermBandMat DComplexHermBandMat::leadingSubmatrix(int k);
Returns the k x k upper left corner of the matrix. The submatrix and the matrix share the same data.
unsigned DComplexHermBandMat::lowerBandwidth();
Returns the lower bandwidth of the matrix.
void DComplexHermBandMat::makeDiagonalReal();
Sets the imaginary part of the main diagonal to 0, thus ensuring that the matrix satisfies the strict mathematical definition of Hermitian.
void DComplexHermBandMat::printOn(ostream&);
Prints the matrix to an output stream in human readable format.
ROCJDComplexRef DComplexHermBandMat::ref(int i, int j);
Returns a reference to the ijth element of the matrix. Bounds checking is done if the preprocessor symbol BOUNDS_CHECK is defined when the header file is read. The member function bcref does the same thing with guaranteed bounds checking.
DComplexHermBandMat DComplexHermBandMat::reference(DComplexHermBandMat&);
Makes this matrix a reference to the argument matrix. The two matrices share the same data. The matrices do not have to be the same size before calling reference. To copy a matrix into another of the same size, use the operator= member operator.
void
DComplexHermBandMat::resize(unsigned n, unsigned n);
void
DComplexHermBandMat::resize(unsigned n,
unsigned n,unsigned hb);
Resizes the matrix or changes both the size of the matrix and the bandwidth. Any new entries in the matrix are set to 0.
void DComplexHermBandMat::restoreFrom(RWFile&);
Reads in a matrix from an RWFile. The matrix must have been stored to the file using the saveOn member function.
void DComplexHermBandMat::restoreFrom(RWvistream&);
Reads in a matrix from an RWvistream, the Rogue Wave virtual input stream class. The matrix must have been stored to the stream using the saveOn member function.
unsigned DComplexHermBandMat::rows();
Returns the number of rows in the matrix.
void DComplexHermBandMat::saveOn(RWFile&);
Stores a matrix to an RWFile. The matrix can be read using the restoreFrom member function.
void DComplexHermBandMat::saveOn(RWvostream&);
Stores a matrix to an RWvostream,the Rogue Wave virtual output stream class. The matrix can be read using the restoreFrom member function.
void DComplexHermBandMat::scanFrom(istream&);
Reads a matrix from an input stream. The format of the matrix is the same as the format output by the printTo member function: first the half bandwidth, followed by the matrix itself. Below is a sample matrix showing the format. Note that extra white space and text preceding the bandwidth specification are ignored. Only the Hermitian part of the relevant band of the matrix is used.
1 5x5 [ (3,0) (7,1) (0,0) (0,0) (0,0) (7,-1) (2,0) (2,-3) (0,0) (0,0) (0,0) (2,3) (8,0) (9,-5) (0,0) (0,0) (0,0) (9,5) (8,0) (7,1) (0,0) (0,0) (0,0) (7,-1) (8,0) ]
void DComplexHermBandMat::set(int i, int j, DComplex x);
Sets the ijth element of the matrix equal to x. Bounds checking is done if the preprocessor symbol BOUNDS_CHECK is defined when the header file is read. The member function bcset does the same thing with guaranteed bounds checking.
unsigned DComplexHermBandMat::upperBandwidth();
Returns the upper bandwidth of the matrix.
DComplex DComplexHermBandMat::val(int i, int j);
Returns the value of the ijth element of the matrix. Bounds checking is done if the preprocessor symbol BOUNDS_CHECK is defined when the header file is read. The member function bcval does the same thing with guaranteed bounds checking.
DComplexHermBandMat DComplexHermBandMat::zero();
Sets every element of the matrix to 0.
ROCJDComplexRef DComplexHermBandMat::operator()(int i,
int j); DComplex DComplexHermBandMat::operator()(int i,
int j)const;
Accesses the ijth element. If the matrix is not a const matrix, a reference type is returned, so this operator can be used for assigning or accessing an element. In this case, using this operator is equivalent to calling the ref member function. If the matrix is a const matrix, a value is returned, so this operator can be used only for accessing an element. In this case, using this operator is equivalent to calling the val member function. Bounds checking is done if the preprocessor symbol BOUNDS_CHECK is defined before including the header file.
DComplexHermBandMat& operator=(const DComplexHermBandMat& A);
Sets the matrix elements equal to the elements of A. The two matrices must be the same size. To make the matrix reference the same data as A, you can use the reference member function.
DComplexHermBandMat& operator==(const DComplexHermBandMat& A); DComplexHermBandMat& operator!=(const DComplexHermBandMat& A);
Boolean operators. Two matrices are considered equal if they have the same size and their elements are all exactly the same. Be aware that floating point arithmetic is not exact; matrices which are theoretically equal are not always numerically equal.
DComplexHermBandMat& operator*=(DComplex x); DComplexHermBandMat& operator/=(DComplex x);
Performs the indicated operation on each element of the matrix.
DComplexHermBandMat& operator+=(const DComplexHermBandMat& A); DComplexHermBandMat& operator-=(const DComplexHermBandMat& A); DComplexHermBandMat& operator*=(const DComplexHermBandMat& A); DComplexHermBandMat& operator/=(const DComplexHermBandMat& A);
Performs element-by-element arithmetic on the data in the matrices. In particular, note that operator*= does element-by-element multiplication, not inner product style matrix multiplication. You can use the product global function to do matrix-matrix inner product multiplication.
DComplexHermBandMat operator+(const DComplexHermBandMat&); DComplexHermBandMat operator-(const DComplexHermBandMat&);
Unary plus and minus operators. Each operator returns a copy of the matrix or its negation.
DComplexHermBandMat operator+(const DComplexHermBandMat&,
const DComplexHermBandMat&); DComplexHermBandMat operator-(const DComplexHermBandMat&,
const DComplexHermBandMat&); DComplexHermBandMat operator*(const DComplexHermBandMat&,
const DComplexHermBandMat&); DComplexHermBandMat operator/(const DComplexHermBandMat&,
const DComplexHermBandMat&);
Performs element-by-element operations on the arguments. To do inner product matrix multiplication, you can use the product global function.
DComplexHermBandMat operator*(DComplex,
const DComplexHermBandMat&); DComplexHermBandMat operator*(const DComplexHermBandMat&,
DComplex); DComplexHermBandMat operator/(const DComplexHermBandMat&,
DComplex);
Performs element-by-element operations on the arguments.
ostream& operator<<(ostream& s, const DComplexHermBandMat&);
Writes the matrix to the stream. This is equivalent to calling the printOn member function.
istream& operator>>(istream& s, const DComplexHermBandMat&);
Reads the matrix from the stream. This is equivalent to calling the scanFrom member function.
DoubleSymBandMat abs(const DComplexHermBandMat& A);
Returns a matrix whose entries are the absolute value of the argument. The absolute value of a complex number is considered to be the sum of the absolute values of its real and imaginary parts. To get the norm of a complex matrix, you can use the norm function.
DComplexHermBandMat conj(const DComplexHermBandMat& A);
Returns a matrix where each element is the complex conjugate of the corresponding element in the matrix A.
DoubleSymMat norm(const DComplexHermBandMat& A);
Returns a matrix where each element is the norm (magnitude) of the corresponding element in the matrix A.
RWMathVec<DComplex> product(const DComplexHermBandMat& A,
const RWMathVec<DComplex>& x);
Returns the inner product (matrix-vector product) of A and x.
RWMathVec<DComplex> product(const RWMathVec<DComplex>& x,
const DComplexHermBandMat& A);
Returns the inner product (matrix-vector product) of x and A. This is equal to the product of A transpose and x.
DoubleSymBandMat real(const DComplexHermBandMat& A);
Returns a matrix where each element is the real part of the corresponding element in the matrix A.
DComplexBandMat toHermBandMat(const RWGenMat<DComplex>&, unsigned h);
Extracts the Hermitian part of a band of entries from a square matrix. The main diagonal is extracted, along with h subdiagonals and superdiagonals. If B is this band, the Hermitian part of the band is (B+conj(BT))/2.
DComplexHermBandMat transpose(const DComplexHermBandMat&);
Returns the transpose of the argument matrix.
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