Toeplitz::ToeplitzSolve Function

Toeplitz Namespace

Overload List

#SignatureDescription
1void ToeplitzSolve(TVec *FirstRow, TVec *FirstCol, TVec *B, TVec *X);Solve a toepltiz system of linear equations.
2void ToeplitzSolve(TVec *FirstRow, TVec *B, TVec *X);Solves toeplitz system.

Overload 1: void ToeplitzSolve(TVec *FirstRow, TVec *FirstCol, TVec *B, TVec *X);

Solve a toepltiz system of linear equations.

#NameTypeDescription
1FirstRowTVec *Defines the first row R(0)..R(N-1) in the equation below. Vector can be real or complex.
2FirstColTVec *Defines the first column C(0)..C(N-2) in the equation below. FirstCol.Length must be equal to FirstRow.Length-1. Vector can be real or complex.
3BTVec *Defines B vector in equation below.
4XTVec *Returns X vector as solution in the equation below.
Remarks:

The procedures solves Toeplitz system of linear equations defined as:

[  R(0)     R(1) ...    R(N-1)  ]       [ X(0)  ]  = [  -B(0)  ]
[  C(0)     R(0)  ...   R(N-2)  ]       [ X(1)  ]  = [  -B(1)  ]
[   ....        .         .     ] x     [   .   ]  = [    .    ]
[  C(N-1) C(N-2) ...    R(1)    ]       [ X(N-2)]  = [ -B(N-2) ]
[  C(N-2) C(N-1) ...    R(0)    ]       [ X(N-1)]  = [ -B(N-1) ]

The computational complexity of the algorithm is O(n2) as oposed to at least O(n3) for solving a general matrix.

Examples
#include "MtxExpr.h"
#include "MtxVec.h"
#include "Units.Toeplitz.h"

using namespace Dew;
using namespace Dew::Math;
using namespace Dew::Math::Units;

void Example()
{

    Vector FirstRow(0);
    Vector FirstCol(0);
    Vector X1(0);
    Vector X2(0);
    Vector B(0);
    Vector R(0);
    Matrix A(0, 0);

    FirstRow.SetIt(false, DewArray<double>{1, 2, 3, 4});
    FirstCol.SetIt(false, DewArray<double>{2, 3, 4});
    A.Toeplitz(FirstRow, FirstCol);
    B.SetIt(false, DewArray<double>{-2, -3, -4, -5});
    R.Copy(FirstRow);
    R.Resize(5);
    R.Values(4) = 5;
    Toeplitz::ToeplitzSolve(FirstRow, FirstCol, B, X1);
    A.LUSolve(B, X2, TMtxType::mtGeneral);
    Toeplitz::Levinson(R, X2);
}
Declared in Dew::Math::Units::Toeplitz · Dew.Math/Units.Toeplitz.h · Cross-compiler

Overload 2: void ToeplitzSolve(TVec *FirstRow, TVec *B, TVec *X);

Solves toeplitz system.

#NameTypeDescription
1FirstRowTVec *Defines the first row R(0)..R(N-1) in the equation below.
2BTVec *Defines B vector in equation below.
3XTVec *Returns X vector in the equation below.
Remarks:

Solves toeplitz system, defined as:

[  R(0)     R(1)* ...   R(N-1)* ]       [ X[1]   ]  = [  -B[1]  ]
[  R(1)     R(0)  ...   R(N-2)* ]       [ X[2]   ]  = [  -B[2]  ]
[   ....        .         .     ]   x   [   .    ]  = [    .    ]
[  R(N-2)   R(N-3) ...   R(1)*  ]       [ X[N-1] ]  = [ -B[N-1] ]
[  R(N-1)   R(N-2) ...   R(0)   ]       [ X[N]   ]  = [ -B[N]   ]
Declared in Dew::Math::Units::Toeplitz · Dew.Math/Units.Toeplitz.h · Cross-compiler