README for Simpson Integration Program

Description

This Java program approximates the integral of the function ( \frac{1}{\sqrt{2\pi}} e^{-x^2/2} ) over the interval ([-1, 1]) using the composite Simpson's 1/3 rule. The program is multithreaded and allows the user to specify the number of threads to use for computation.

Requirements

• Java Development Kit (JDK) installed on your system.
• A terminal or command prompt to compile and run the program.

How to Compile and Run

1. Save the Program:

   • Save the program in a file named SimpsonIntegration.java.

2. Compile the Program:

   • Open a terminal or command prompt and navigate to the directory where the program is saved.
   • Compile the program using the following command:

     javac SimpsonIntegration.java

3. Run the Program:

   • Run the program by specifying the number of threads as a command-line argument. For example, to use 8 threads:

     java SimpsonIntegration 8

Command-Line Arguments

• The program requires one command-line argument:
  • <number_of_threads>: The number of threads to use for computation (e.g., 4, 8, 16).

Output

• The program will output the approximate value of the integral computed using the specified number of threads and the time taken to do so.

Example

javac SimpsonIntegration.java
java SimpsonIntegration 8

Output:

Integral value: 0.6826894921370761
Execution time: 50.3498 milliseconds

Notes

• The program uses the composite Simpson's 1/3 rule for numerical integration.
• The number of points used for integration is fixed at ( 10^6 ).
• The program demonstrates how multithreading can be used to parallelize numerical computations.

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README for Matrix Multiplication Program

Description

This Java program performs matrix multiplication ( C = A \times B ) for two square matrices ( A ) and ( B ) of size ( 1000 \times 1000 ). The matrices are initialized with random values between 0 and 10. The program is multithreaded and allows the user to specify the number of threads to use for computation.

Requirements

• Java Development Kit (JDK) installed on your system.
• A terminal or command prompt to compile and run the program.

How to Compile and Run

1. Save the Program:

   • Save the program in a file named MatrixMultiplication.java.

2. Compile the Program:

   • Open a terminal or command prompt and navigate to the directory where the program is saved.
   • Compile the program using the following command:

     javac MatrixMultiplication.java

3. Run the Program:

   • Run the program by specifying the number of threads as a command-line argument. For example, to use 8 threads:

     java MatrixMultiplication 8

Command-Line Arguments

• The program requires one command-line argument:
  • <number_of_threads>: The number of threads to use for computation (e.g., 8, 10, 50, 100, 500).

Output

• The program will print the time of execution of code.

Example

javac MatrixMultiplication.java
java MatrixMultiplication 8

Output:

Matrix multiplication completed.
Execution time: 844.9204 milliseconds

Notes

• The program demonstrates how multithreading can be used to parallelize matrix multiplication.
• The matrices are initialized with random values between 0 and 10.
• Only the first ( 10 \times 10 ) portion of the matrices is printed to avoid overwhelming the console with output.

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