Untitled Quiz

First Numerical Problem:

• Presented by the MAP Department at EARIST Manila.
• Includes topics on Quadratic Equations, 1st Order Differential Equations, Euler's Method, File and File Processing, and Radioactive Decay.

Content:

• Quadratic Equation
• 1st Order Differential Equation
• Euler's Method
• Introduction To Files And File Processing
• Radioactive Decay

Quadratic Equation:

• Derived from the Latin word "quadratus" meaning "square".
• Represents an equation of the form ax² + bx + c = 0, where a, b, and c are known numbers and a ≠ 0.
• Has two solutions that can be real or complex, and may or may not be distinct.
• Solution formulas:
  • x = (-b ± √(b² - 4ac)) / 2a
  • y = ax² + bx +c
  • Other methods include factoring, completing the square.

Solving the Quadratic Equation:

• Factoring by inspection (px + q)(rx + s) = 0
• Completing the square. x² + 2hx + h² = (x +h)²
• Quadratic formula: x=(-b±√b²-4ac)/2a

Avoiding Loss of Significance:

• Lines 19-22 in the program use MAX to find the largest coefficient value.
• Scaling coefficients speeds up calculations.
• Line 29 in the program is for roots that are very close together.
• Scaling coefficients approximates, sacrificing range and precision.
• The presented code functions when all coefficients are in the same magnitude, or a << b.

1st Order ODE:

• Differential equations involving derivatives of one or more functions.

Differential Equations:

• Equations involving derivatives of one or more functions.
• Examples are shown as:
• dy/dx = 6x, for instance.
• Initial information provides how a system changes, determining the function representing that change.

Derivative Notations:

• Leibniz, Lagrange, and Newton notations for derivatives are presented.

Order of Differential Equation:

• The highest order of derivative present in the equation.
• Examples listed are 1, 2, 3, and 4.
• Examples presented are 1, 2, 3, 4, 5, and 6.

Solution of a Differential Equation:

• A function satisfying the equation over some interval.
• Function's first derivative must be continuous on the given interval.

Ordinary Differential Equations:

• Equations involving derivatives with respect to a single independent variable.
• Examples given are 1, 2, and 3.

Solution to a Differential Equation:

• Steps to verify example solution:
  • Solve the equation.
  • Apply chain rule.
  • Plug the solution into the equation to confirm.

Solution of a Differential Equation (Question):

• How many solutions exist for the equation?
• Steps to solve include:
  • Separate the variables.
  • Integrate both sides of the equation.

Particular Solution:

• Solving a differential equation by assigning specific values to arbitrary constants within the general solution.
• This solution process is done by inserting the given details into the general solution.
• Method of solving particular solutions steps include:
  • Insert initial condition into the equation.
  • Solve for the values

Direct Integration:

• Simplifying differential equations that have a right-hand expression involving only the independent variable (x) and not the dependent variable (y).
• The solution involves integrating both sides of the equation with respect to x to solve.

Example (1-3): Direct Integration:

• Instructions/Solution steps to solve the given differential equations are presented.

1st Order ODE:

• Overview of 1st-order ordinary differential equations.

Types Of Linear DE:

• Separable Variables, Homogeneous Equation, Exact Equation, and Linear Equation.

Variable Separable Method:

• The general form of variable separable method equations.
• This type of equation can be solved by direct integration.

Homogeneous Equations:

• A function f(x, y) is homogeneous of degree n in x and y if f(tx, ty) = tⁿf(x, y). A first-order DE M(x, y) dx + N(x, y) dy = 0 is called homogeneous if M(x, y) and N(x, y) are homogeneous functions of the same degree in x and y. This can be solved by substitution y = vx.

Example Homogeneous Equations:

• Show how to solve −2xy dx.

Euler's Method:

• Procedure for approximating solutions to differential equations.

Local Linearity and Approximation:

• Approximates a function using local linearity, allowing for repetitive calculations.

Implementing the Euler's Method:

• Necessary components for Euler's method.
  • Differential equation expressed as y' = some expression in t and y.
  • A point (t₀, y₀) on the solution graph y = f(t).
  • A fixed step size, ∆t

Example Euler's Method Calculation:

• The procedure of solving the differential equation example via Euler's Method, in table format, given y' = sin(t²), (1,1) lie on graph, ∆t = 0.1

Using Euler's Method:

• Examples show calculation of y(0.3) using a step size of 0.1 for a given initial value problem.

Introduction to Files and File Processing:

• Defining a file as a unit of data held outside computer memory.
• Files contain records (sequential components) accessible with READ and WRITE commands.
• Secondary memory is the area where unneeded programs and data are stored.

Input/Output Unit:

• A number or asterisk that refers to an external or internal unit in input/output statements.

Number and Asterisk:

• Used to refer to an external file unit that can be connected and disconnected using OPEN and CLOSE statements.
• Asterisk refers to standard input/output devices.

File Name:

• Refers to internal file units where it is a character variable in program memory; integer value is mandatory.

Example Programs (Several Files):

• Example programs to read and write to files with details on open and close statements.

Fortran Input/Output Statements:

• List and descriptions of frequently used Fortran i/o statements (open, close, read, write, rewind, backspace).

The Open Statement:

• Format (OPEN(open_list)). Clauses specify i/o unit numbers, file name, and access method.
• Clause details listed in subsequent sections.

Clauses for the OPEN Statement:

• Details of clauses for the OPEN statement: UNIT, FILE, STATUS, ACTION,and IOSTAT

CASE 1: Opening a File for Input:

• Example Fortran code to open a file for input and the significance of the various parameters

File Processing OPEN (open-list):

• Procedures for opening, and specifying the status and action of the files.

OPEN (open-list) Cont:

• Description of different parameters and their uses to process files within the OPEN statement: ACTION, POSITION, and IOSTAT

###Examples:

• Several examples on how to use OPEN statement to open files, along with variable names to hold file specific parameters.

File Processing Close (close-list):

• Includes directives to specify a unit number and IOSTAT clauses to process files

Write (control-list) output-list:

• Provides functions for formatting the output of data.
  • Describes the unit specifier, and how to specify the output format, advance control, and other useful file processing commands

Read (control-list) input-list and Examples:

• Functions for input-list from a data source.
• Format descriptions of common READ statements.

File Input/Output:

• Detailed information on accessing files for input/output, using READ and WRITE.
• How to check and handle end-of-file conditions (EOF) (using IOSTAT) and input errors.

Example:

• Example Fortran program to read from a file (with descriptions of variables) and handle possible errors

Rewind/Backspace:

• Commands for repositioning file: rewind (start), backspace (previous lines).