UTK - Math 171 - Alexiades

            How-To's

numbers

  • Enter very big and very small numbers in scientific notation: e.g. 3.14e6 , 3.14e-6
  • Print very big and very small numbers with '%e' formating.

    Expressions
  • Guard against subtraction of nearly equal numbers, multiplying by very small number (or dividing by very large number).
      Try to re-write the expression somehow.
  • The quadratic formula is subject to loss of significant digits when   0 < 4ac << b2 (much less than)
      A safe way to find roots of a quadratic:   x1 = ( −b−sign(b)√(b2−4ac) ) / 2a ,   x2 = c / ax1
  • Always evaluate polynomials in nested form: saves operations and may reduce roundoff.
      e.g. ax2 + bx + c = c + x*(b+a*x) ,   e.g. ax3 + bx2 + cx + d = d + x*(c+x*(b+a*x)) , etc

    Matlab
  • All variables inside a function are local to the function. Only output variables will return to calling program.
  • Variables in a script (and workspace) are global, can be passed to functions called by the script.
  • A = rand(m,n) : m×n matrix of random numbers drawn uniformly from (0 , 1)
  • If x is an array: [m, imin] = min(x) gives smallest element m and the index imin where it occurs.
  • If x is an array: [M, imax] = max(x) gives largest element M and the index imax where it occurs.
  • If A is a matrix: A(:) is 1D array of A by columns. r3=A(3,:) is 3rd row of A,   c3=A(:,3) is 3rd column of A.
  • If A is a matrix: [m, imin] = min( A(:) )   gives min element of A and its index in the array A(:).
  • Preallocate arrays (before entering loops): a = zeros(1,N);   b = zeros(M,1);   A = zeros(M,N);
  • Calculation with arrays is faster than via loops on components, especially in Matlab.
      Caution: sum, min, max on a matrix A operate on each column of A. Use A(:) = col array of columns of A.

  • Finding Sum, Min, Max of matrix of size M×N using nested loops: 
    	M = input('enter M: ')
	N = input('enter N: ')
	A = 10*rand(M,N);
	%----using array ops: on A(:), else they apply on each column
	arraySum = sum( A(:) );   fprintf(' arraySum= %f \n', arraySum )
	[arrayMin, iarrayMin] = min( A(:) )
	     fprintf(' arrayMin= %f at i: %d \n', arrayMin, iarrayMin )
	[arrayMax, iarrayMax] = max( A(:) )
	     fprintf(' arrayMax= %f at i: %d \n', arrayMax, iarrayMax )
	
	%----using loops:
	Sum = 0 ;  Min = Inf ;  Max = -Inf ;
	for i = 1:M		% for each row i
	   for j = 1:N		% for each col j
		aij = A(i,j);	%save to avoid multiple lookups
			Sum = Sum + aij;
		if( aij <= Min )
                	Min = aij;   iMin = i; jMin = j;
        	end %if
        	if( aij >= Max )
                	Max = aij;   iMax = i; jMax = j;
        	end %if
	   end %j
	end %i
	fprintf(' Sum= %f \n', Sum)
	fprintf(' Min= %f at i,j: %d %d \n', Min, iMin, jMin )
	fprintf(' Max= %f at i,j: %d %d \n', Max, iMax, jMax )
  • important Matlab built-in functions:
      input   linspace   rand   randi   plot   fopen,fclose   fprintf   fscanf   fgetl   feof
      zeros   ones   length   size   sort   sum   min   max   norm  

  • Matlab Workbook for practicing basics