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Matrices and Vectors

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• Problem Description
  • We will student how to provide the following functionalities

    Matrix * Vector multiplication Vector * Matrix multiplication

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• The Matrix3x3 and Vector3 Classes

  • I will again use 3x3 matrices to learn how to design these functionalities to focus on programming issues rather than algorithmic issues

  • The Matrix3x3 class:

    class Matrix3x3 { public: double A[3][3]; }; // <--- Note: class definition must end in ;

  • The Vector3 class:

    class Vector3 { public: double d[3]; }; // <--- Note: class definition must end in ;

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• Matrix * Vector multiplication
  • To compute one element in a Matrix * Vector product, we must add a number of products :

    +- -+ +- -+ +- -+ | 1 0 0 | | 3 | | .. | | 1 1 0 | * | 6 | = | .. | | 1 2 5 | | 9 | | 1*3 + 2*6 + 5*9 | +- -+ +- -+ +- -+ A * v = u

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  • In the example above, we see that:

    u[2] = a[2][0]*v[0] + a[2][1]*v[1] + a[2][2]*v[2]

  • This for loop is used to compute the element u[2] in a 3x3 matrix-vector product:

    u[2] = 0; for (k = 0; k < 3; k = k + 1) u[2] = u[2] + A[2][k] * v[k];

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  • In general, the following for loop is used to compute the element u[i] in a matrix-vector product:

    u[i] = 0.0; for (k = 0; k < N; k = k + 1) u[i] = u[i] + A[i][k] * v[k];

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  • To compute every element v[i], the above loop is executed for every element v[i] by using a single for-loop:

    for (i = 0; i < N; i = i + 1) { u[i] = 0.0; for (k = 0; k < N; k = k + 1) u[i] = u[i] + A[i][k] * v[k]; }

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• A Matrix * Vector multiply operator
  • To provide the functionality

    Matrix3x3 M; Vector3 u, v; u = M * v;

    we must define the following operator function :

    Vector3 operator* ( Matrix3x3 & M, Vector3 & v) { .... }

    (The body of the function must implement the matrix-vector multiplication algorithm )

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  • Matrix*Vector multply algorithm:

    Vector3 operator* ( Matrix3x3 & M, Vector3 & v) { Vector3 out; for (i = 0; i < 3; i = i + 1) { out.d[i] = 0.0; for (k = 0; k < 3; k = k + 1) out.d[i] = out.d[i] + M.A[i][k] * v.d[k]; } return(out); }

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  • How to use this function:

    int main(int argc, char *argv[]) { Matrix3x3 A; Vector3 v, u; u = A * v; }

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  • Example Program: (Demo above code)                                                
    • Prog file: click here

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  • Member function implementation:
    • You can provide this functionality through a member function

    • Because the first parameter is of the type Matrix3x3 , the member function must be define in the Matrix3x3 class

    class Matrix3x3 { public: double A[3][3]; Vector3 operator* ( Vector3 & v) { Vector3 out; for (i = 0; i < 3; i = i + 1) { out.d[i] = 0.0; for (k = 0; k < 3; k = k + 1) out.d[i] = out.d[i] + A[i][k] * v.d[k]; } return(out); } };

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  • Example Program: (Demo above code)                                                
    • Prog file: click here

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• A Vector * Matrix multiply operator
  • To provide the functionality

    Matrix3x3 M; Vector3 u, v; u = v * M; // Vector * Matrix !!

    we must define the following operator function :

    // Notice that the order of parameter is changed !!! Vector3 operator* ( Vector3 & v, Matrix3x3 & M ) { .... }

    (We must write a new function to implement the vector-matrix multiplication algorithm )

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  • Vector*Matrix multply algorithm:

    Vector3 operator* ( Vector3 & v, Matrix3x3 & M ) { Vector3 out; for (i = 0; i < 3; i = i + 1) { out.d[i] = 0.0; for (k = 0; k < 3; k = k + 1) out.d[i] = out.d[i] + v.d[k] * M.A[k][i]; } return(out); }

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  • How to use this function:

    int main(int argc, char *argv[]) { Matrix3x3 A; Vector3 v, u; u = v * A; }

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  • Example Program: (Demo above code)                                                
    • Prog file: click here

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  • Member function implementation:
    • You can provide this functionality through a member function

    • Because the first parameter is of the type Vector3 , the member function must be define in the Vector3 class

    class Vector3 { public: double A[3][3]; Vector3 operator* ( Matrix3x3 & M ) { Vector3 out; for (i = 0; i < 3; i = i + 1) { out.d[i] = 0.0; for (k = 0; k < 3; k = k + 1) out.d[i] = out.d[i] + d[k] * M.A[k][i]; } return(out); } };

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  • Example Program: (Demo above code)                                                
    • Prog file: click here

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• Advantage of using member functions
  • Member functions has special access prilileges to the member variables of the same class

    Using member functions will allow you to improve on correctness of the entire problem by localizing program statements that can cause errors

  • Using the polymorphism mechanism that is only available through inheritance, programs using member functions can be re-directed to use other implementations of that member function ,