NUMERICAL METHODS WITH MATLAB PROGRAMMING

NUMERICAL METHODS WITH MATLAB PROGRAMMING

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  NUMERICAL METHODS WITH MATLABPROGRAMMING By A.DHARANI ASSISTANT PROFFESSOR PG DEPARTMENT OF MATHEMATICS SRI SARADA NIKETAN COLLEGE FOR WOMEN
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  STATEMENT LEVEL CONTROL STRUCTURES One way to think of computer programs is to consider how the statements that compose the program are organized. Usually, sections of computer code can be categorized into on of three structures: sequences, selection structures and repetition structures.  Sequences: Sequences are lists of commands that are executed one after another.  Selection structure: A selection structure allows the programmer to execute one command or group of commands if some criteria is true and a second set of commands if the criteria is false. A selection statement provides the means of choosing between these paths based on a logical condition. The conditions that are evaluated often contain relational and logical operators o r functions.  Repetition structure: A repetition structure or loop, causes a group of statements to be executed zero, one, or more times. The number of times a loop is executed depends on either a counter or the evaluation of a logical condition.
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  RELATIONAL OPERATORS Relational OperatorsInterpretation < Less than <= Less than or equal to > Greater than >= Greater than or equal to == Equal to ~= Not equal to
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  EXAMPLES  MATLAB hassix relational operators for comparing two matrices of equal size, comparisons are either true or false and most computer programs use the number 1 for true and 0 for false.  x = [ 1 2 3 4 5 ];  y = [-2 0 2 4 6];  x<y  returns  ans=  0 0 0 0 1  which tells that the comparisons was false for the first four elements but true for the last one.
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  MATLAB ALSO ALLOWSUS TO COMPARE COMPARISONS WITH LOGICAL OPERATORS: AND, NOT, AND OR Logical Operators interpretation & and ~ not | or
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  EXAMPLE FOR LOGICAL OPERATORS considerthe following: x = [ 1 2 3 4 5]; y = [ -2 0 2 4 6]; z = [ 8 8 8 8 8 ]; z>x & z>y returns ans= 1 1 1 1 1 Both z is greater than both x and y for every element.
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  • Selection structuresMATLAB offers two kinds of selection structures: find and a family of if structures • Find The find command is unique to MATLAB, and can often be used instead of both if and loop structures. The usefulness of the find command is best described with examples. Assume that you have a list of temperatures measured in a manufacturing process. If the temperature is less than 95 degree F, them widgets produced will be faulty: temp = [ 100 98 94 101 93 ] use the find function to determine which widgets are family: find(temp<95) Returns a vector of element numbers; ans= 3 5
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  Location of allelements greater than 9: x = [ 1,2,3 10,5,1 3,12,2 8,3,1] element = find(x>9) [row, column] = find(x>9) Returns x = 1 2 3 10 5 1 3 12 2 8 3 1 Element = 2 7 row = 2 3 column = 1 2 The numbers 10 and 12 are the only two values greater than 9
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  IF , ELSEIF, ELSE: EXAMPLE: a = input (‘ enter a = ‘); if a>0 disp (‘ a is positive’) elseif a<0 disp (‘ a is negative’) else disp (‘ a is zero’) end Returns, ‘enter a = 5’ ans = 5 is positive ‘ enter a = -1’ ans = -1 is negative ‘ enter a = 0’ ans = zero
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  IF: If the logicalexpression is true, the statement between if and end are executed. If the logical expression is false, program control jumps immediately to the statement following the end statement. ELSE IF: It is a conditional statement performed after an if statement that, if true, performs a function. The elseif statement is only executed if the preceding if expression evaluated to false, and the current elseif expression evaluated to true. ELSE: Use else to specify a block of code to be executed, if the same condition is false. Else doesn’t need a condition as it is the default for everything where as else if is still an if so it needs a condition.
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  . • Loops Aloop is a structure that allows you to repeat a set of statements. In general, you should avoid loops in MATLAB because they are seldom needed, and they can significantly increase the execution time of a program. For loops In general, it Is possible to use either a for or a while loop in any situation that requires a repetition structures. However, for loops are the easier choice when you know how many times you want to repeat a set of instructions. The general format is for index = expression statements end Usually, the expression is a vector, and the statements are repeated as many times as there are columns in the expression matrix. For example, to find 5 raised to the 100th power, first initialize a running total: total = 1; for k = 1:100 total = total*5; end
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  The first timethrough the loop, total = 1, so total ×5=5, the next line through the loop, the value of total is updated to 5×5 equals 25,after 100 times through the loop, the final value is found, and corresponds to 5ˆ100. Notice that the value of totalwas suppressed, so it won’t print out each time through the loop. To recall the find value of total, type total which returns total = 7.8886e+069 Another example of loop is, for k = 1:4 k end returns k = 1 k = 2 k= 3 k = 4
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  THE RULES FORWRITING AND USING A FOR LOOP ARE THE FOLLOWING: • The index of a for loop must be a variable. Although k is often used as the symbol for the index, any variable name can be used. The use of k is strictly a style issue. •If the expression matrix is the empty matrix, the loop will not be executed. Control will pass to the statement following the end statement. • If the expression matrix is a scalar, the loop will be executed one time, with the index containing the value of the scalar. • If the expression is a row vector, each time through the loop the index will contain the next column in the matrix. • If the expression matrix, each time through the loop the index will contain the next column in the matrix. This means that the index will be a column vector. • Upon completion of a for loop, the index contains the last value used. The colon operator can be used to define the expression matrix using the following format: for k = initial : increment : limit
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  • While loops: Thewhile loop is a structure used for repeating a set of statements as long as specified condition is true. The general format for this control structure is while expression Statements end The statements in the while loop are executed as long as the real part of the expression has all nonzero elements. The expression is usually a comparison using relational and logical operators. When the result of a comparison is true, the result is 1, and therefore ‘nonzero’. The loop will continue repeating as long as the comparison is still true. When the expression is evaluated as false, control skips to the statement following the end statement . Consider the following example: First initialize a: a = 0; Then find the smallest multiple of 3 that is less than 100:
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  while( a <100 ) a = a+3; end; The last time through the loop a will start out as 99, then will become 102 when 3 is added to 99. The smallest multiple then becomes a – 3 Which returns ans = 99 The variable modified in the statements inside the loop should include the variables in the expression, or else the value of the expression will never change. If the expression is always true, then the loop will execute an infinite number of times.
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  COMMANDS AND FUNCTIONS COMMANDMEANINGS clock Determines the current time on the CPU clock disp Displays matrix or test else Defines the path if the result of an if statement is false elseif Defines the path if the result of an if statement is false, and specifies a new logical test end Identifies the end of a control structure etime Finds elapsed time find Determines which elements in a matrix meet the input criteria for Generates a loop structure fprintf Prints formatted information function Identifies an M-files as a function if Tests a logical expression
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  COMMAND MEANING input Promptsthe user to enter a value meshgrid Maps two input vectors onto two 2-D matrices nargin Determines the number of input arguments in a function nargout Determines the number of output arguments from a function num2string Converts an array into a string ones Creates a matrix of ones tic Starts a timing sequence toc Stops a timing sequence while Generates a loop structure zeros Creates a matrix of zeros
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  THANK YOU