Java in Science: Data Inter- and Extrapolation Using Numerical Methods of Polynomial Fittings, Part 2

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Review Part 1

Sums and Scalar Multiples

Two matrices would be equal if they have the same size (that
is, the same number of rows and the same number of columns) and also their
corresponding entries are equal. Matrix A is not equal
to matrix B, although they have the same matrix
size = 3 x 2. The corresponding entries at index (1,1)
are not the same for matrix A and B because
A(1,1) = 3 and B(1,1) = 4, and so they are
different. Note that the entry-index starts at zero.

double[][] arrayA = {{4.,0.},{5.,3.},{-2.,1.}};

//Instantiate matrix object A

Matrix A = new Matrix(arrayA);

double[][] arrayB = {{4.,0.},{5.,4.},{-2.,1.}};

//Instantiate matrix object B

Matrix B = new Matrix(arrayB);

double[][] arrayC = {{1.,7.},{7.,4.}};

//Instantiate matrix object C

Matrix C = Matrix(arrayC);

double[][] arrayD = {{1.,0.},{0.,1.}};

//Instantiate matrix object D

Matrix D = new Matrix(arrayD);

The additions (sums) and subtractions (difference) of matrices is done
element-wise. The sum of A+B would result in a
matrix the same size as A or B with all
entries as the sum of a_ij and b_ij
, for example, the resultant entry of (1,1) = 7,
which is a₁₁+b₁₁ = 3+4 = 7.
The sum of two different size (dimemsion) matrices is undefined such as
A+C, because size(A) = 3 by 2
and size(C) = 2 by 2. In multiplying a matrix by
a scaler results in a matrix with all
the entries scaled by this scaler-value. 3*B is a
multiplication of B by a scaler value of 3 in
which all entries are now 3b_ij.

Matrix A_plus_B = A.plus(B) ;

//returned matrix has all its entries scaled by 3.

Matrix matrix_3_times_B = B.times(3.);

//This following line will throw an IllegalArgumentException because matrix dimensions do

//not agree, matrix A is a 4 by 2 but C is a 2 by 2 size matrix.

Matrix A_plus_C = A.plus(C);

Matrix Multiplication

Under matrix multiplication, such as A*C, the inner
dimensions have to be the same for it to be defined, regardless of the outer
dimensions. In this case size(A) = [3 x 2] and size(C)
= [2 x 2]; therefore, matrix multiplication is defined because they have the
same inner dimensions of two, [3 x 2][2 x 2].
The dimension of the result of multiplying two matrices would be the outer
dimensions of the respective matrices. A*C would
result in a matrix with size(A*C) = [3 x 2],
that is [3 x 2][2 x 2].
Matrix multiplication results in all entries(i,j) equals
the summation of element-wise multiplication of each row of A
to each column of C. Multiplication of A
*B is undefined, because the inner dimensions do not agree:
A*
B is [3 x 2]*[3 x 2]. Multiplication of
matrices is not always commutative (order of operation), B
*C is defined, but reversing the order as
C*B is undefined but C*
D = D*C. Matrix multiplication is directly
in contrast with its elementary counterpart, 3 x 4 =
4 x 3 = 12.
Matrix D is a special type called Identity, and this
is the equivalent in elementary arithmetic of the number 1,
which is the multiplication identity, 6 x 1
= 6. Identity is
a square-matrix (number of rows = columns) with any
dimension and all its elements at the diagonal, from
top-left to bottom-right are ones and zeros everywhere. Any matrix
that is multiplied to the Identity is always that matrix, examples are:
A*D = A, B*
D = B and C*D =
C.

Matrix A_times_C = A.times(C) ;

//The following line will throw an IllegalArgumentException because inner matrix

//dimensions do not agree.

Matrix A_times_B = A.times(B);

Transpose of a Matrix

Given an m x n matrix E, the
transpose of E is the n x m
matrix denoted by E^T, whose columns are formed from
the corresponding rows of E. General matrix transposition
operations:

1. (F + G)^T =
F^T + G^T

Matrix result1 = F.plus(G).transpose() ;

Matrix result2 = F.transpose().plus(G.transpose());

//Matrix result1 and result2 are equals (same value in all entries)

2. (F^T)^T = F

Matrix result3 = F.transpose().transpose() ;

//Matrix result3 and F are equals (same value in all entries)

3. (F * G)^T =
G^T * F^T

Matrix result4 = F.times(G).transpose() ;

Matrix result5 = G.transpose().times(F.transpose()) ;

//Matrix result4 and result5 are equals (same value in all entries)

4. For any scalar r, (r *
F)^T = r * F^T

Matrix result6 = F.times(r).transpose() ;

Matrix result7 = F.transpose().times(r) ;

//Matrix result6 and result7 are equals (same value in all entries)

Inverse of a Matrix

If K is an [n x n] matrix,
sometimes there is another [n x n] matrix
L such that:

K * L = I and L *
K = I
where I is an [n x
n] Identity matrix

We say K is invertible and L is an inverse of K.
The inverse of K is denoted by K^-1.

K * K^-1 = I, where I –
is the Identity matrix
K^-1 * K = I

Matrix inverseMatrix = K.inverse() ;

//Matrix inverseMatrix is an inverse of matrix K

QR Matrix Factorization

Factorizing a matrix follows a similar concept in elementary arithmetics. The
number 30 has prime factors of = 2 x 3
x 5. If P is an [m x n]
matrix with linearly independent columns, then it can be factored as:

P = Q * R:
where Q is an [m x m] whose columns form
an orthonormal basis and R is an [m x n]
upper triangular invertible matrix with positive entries on its diagonal.

double[][] arrayP = {{1.,0.},{1.,1.},{1.,1.},{1.,1.}};

Matrix P = new Matrix(arrayP);

//Create a QR factorisation object from matrix P.

QRDecomposition QR = new QRDecomposition(P);

Matrix Q = QR.getQ();

Matrix R = QR.getR();

//Now that we have two matrix factors for matrix P which:  Q*R = P

Elementary Matrix Function and Special Matrices

double[][] arrayA = {{7,5},{2,3}};

Matrix A = new Matrix(arrayA);

//Create matrix object B by tiling matrix A in a 3 by 2 pattern.

Matrix B = JElmat.repmat(A,3,2);

//Create Vandermonde-matrix

double[] arrayA = {5,-1,7,6};

//vander is also overloaded with one input argument a double[] array, where columns = rows

Matrix A = JElmat.vander(arrayA);

double[] arrayB = {5,-1,7,6,-3,4};

//Create a Vandermonde-matrix with only four columns.

Matrix B = JElmat.vander(arrayB,4);

double arrayA = {{19,18,16,18},{5,15,9,15},{12,9,12,4},{10,0,16,8}};

Matrix A = new Matrix(arrayA);

//Sum all the columns of matrix A.

Matrix B = JDatafun.sum(A);

//row matrix A, starting at -8.	and ending at 12. with eleven points equally spaced.

//internal array A = (-8    -6    -4    -2     0     2     4     6     8    10    12)

Matrix A = JElmat.linspace(-8.,12.,11);

double[][] arrayA = {{1.,2.,3.,4.},{5.,6.,7.,8.},{9.,10.,11.,12.}};

Matrix A = new Matrix(arrayA);

//reshape matrix A from 3 x 4 dimension into 6 x 2

Matrix B = JElmat.reshape(A,6,2);

//Create a matrix of ones with 3 x 4 size.

Matrix X = JElmat.ones(3,4);

//Create a matrix of zeros with 2 x 2 size.

Matrix Y = JElmat.zeros(2,2);

We will continue with polynomial fittings and Java code in Part 3.

Download Jama and
Jamlab and
Jamlab documentation.

References

• Java for Engineers and Scientists by Stephen J. Chapman, Prentice Hall, 1999.
• Introductory Java for Scientists and Engineers by Richard J. Davies, Addison-Wesley Pub. Co., 1999.
• Applied Numerical Analysis (Sixth Edition) by Curtis F. Gerald and Patrick O. Wheatly, Addison-Wesley Pub. Co., 1999.
• Linear Algebra and Its Applications (Second Edition), by David C. Lay, Addison-Wesley Pub. Co.
• Numerical Recipes in Fortran 77, the Art of Scientific Computing (Volume 1) by William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery, Cambridge University Press, 1997.
• Mastering MATLAB 5: A Comprehensive Tutorial and Reference by Duane Hanselman and Bruce Littlefield, Prentice Hall, 1997.
• Advanced Mathematics and Mechanics Applications using MATLAB by Louis H. Turcotte and Howard B. Wilson, CRC Press, 1998.

Related Articles

• Java as a Scientific Programming Language (Part 1): More Issues for Scientific Programming in Java
• Scientific Computing in Java (Part 2): Writing Scientific Programs in Java
• Using Java in Scientific Research: An Introduction

About the Author

Sione Palu is a Java developer at Datacom in Auckland, New Zealand, currently involved in a Web application development project. Palu graduated from the University of Auckland, New Zealand, double majoring in mathematics and computer science. He has a personal interest in applying Java and mathematics in the fields of mathematical modeling and simulations, expert systems, neural and soft computation, wavelets, digital signal processing, and control systems.