Numerical Methods-Assignment-2
Numerical errors, Error Propagation, Basic Concepts of Iteration & Stability
(1) Let Mbe a sufficiently large number which results in an overflow of memory and mbe a suffi-
ciently small number which results in underflow of memory of a computing device. Then give the
output of the following operations:
(i) M×m(ii) M/m(iii) m/m(iv) k×M(v) k×m(vi) m/M, where kis a constant.
(2) In the following problems, show all the steps involved in the computation.
(i) Using 5−digit rounding and chopping, compute 37654 +25.874 −37679.
(ii) Let a=0.00456, b=0.123, c=0.128. Using 3-digit rounding and chopping, compute
(a+b) + cand a+ (b+c). What is your conclusion?
(iii) Let a=2, b=0.6, c=0.602. Using 3−digit rounding and chopping, compute (a×b) + (a×
c). What is your conclusion?
(3) Let x,y, and zbe real numbers whose floating point approximations in a computing device co-
incide with x,y, and z, respectively. Show that the relative error in computing x(y+z)equals
ε1+ε2−ε1ε2, where ε1=Er(f l(y+z)) and ε2=Er(f l (x×f l(y+z))).
(4) Let ε=Er(f l(x)). Show that
(i)|ε| ≤ 10−n+1if the computing device uses n−digit (decimal) chopping.
(ii) |ε| ≤ 1
210−n+1if the computing device uses n−digit (decimal) rounding.
(iii) Can the equality hold in the above inequalities?
(5) Let xA=3.14 and yA=2.651 be obtained from the numbers xTand yTrespectively using 4-digit
rounding. For any such values of xTand yT, find the smallest interval that contains
(i) xT+yT(ii) xT/yT(iii) xT×yT.
(6) Let x<0<ybe such that the approximate numbers xAand yAhas seven and nine significant digits
with xand yrespectively. Show that zA:=xA−yAhas at least six significant digits when compared
to z:=x−y.
(7) Let xTbe a real number. Let xA=2.5 be an approximate value of xTwith an absolute error at
most 0.01. The function f(x) = x3is evaluated at x=xAinstead of x=xT. Estimate the resulting
absolute error.
(8) Let f:R→Rand g:R→Rbe continuously differentiable functions such that
(i) there exists constant M>0 such that |f0(x)| ≥ Mand |g0(x)| ≤ Mfor all x∈R,
(ii) the condition number of fis less than 1, and
(iii) the condition number of gis greater than 1.
Show that |g(x)|<|f(x)|for all x∈R.
(9) Find the condition number at a point x=cfor the following functions
(i) f(x) = x2, (ii) g(x) = πx, (iii) h(x) = bx.
(10) check for stability of computing the function h(x) = sin2(x)
1−cos2xfor values of xvery close to 0. Run
in Matlab and see the output for x=10(−7.97)for the function h(x)and also try with some non-
negative number less than x, for instance, x=10−8. Note that for any x∈R,h(x) = 1.
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