ALL 9 CHAPTERS COVERED
SOLUTIONS MANUAL
,Solutions Manual for Chapter 1
1.1 Program:
Defining Constants
fraction = 0.2; %The decimal fraction to be converted to binary
digits = 16; % Number of binary digits in the binary number to be determined
binary(1:digits)=0; %The binary significand
Determining the binary significand
for i = 1:digits
base2power = (1/2)^i;
if (fraction >= base2power)
fraction = fraction - base2power;
binary(i) = 1;
else
binary(i) = 0;
end
end
16-digit binary significand for 0.7 is 0.1011 0011 0011 0011
16-digit binary significand for 0.2 is 0.0011 0011 0011 0011
1.2 (i) (a) 0. 0 0 0 1.
(b) 0. 1 0 1 0
(ii) 0. 1 0 1 1
(iii) 0.6875
(iv) relative error: 11.61%
1.3 (a) 1.1001 × 2110
(b) 1.010000000010101100000010000011000100100110111010011 × 2110
(c) 1.0000 × 2-110.
1
1.4 relative error using 6 significant digits is 0.2 or 20%. Instead by evaluating ,
x + 1 + x2
2
we get a relative error of -0.000203 or -0.0203%. The rearrangement reduced the error by 1000
times or 3 orders of magnitude!
1.5 The first expression gives us 7.1 and the second expression gives us 7.106. The second
expression produces a more accurate result. The loss of accuracy is due to the multiplication
,steps, where we lose information due to 4-digit rounding arithmetic. The first expression
involves 7 multiplication steps. The 2nd expression involves only 2 multiplication steps.
1.6 Absolute error is 0.0076. After summing the smallest terms first, and in the order of smallest
to largest numbers, maintaining 3-digit rounding, the absolute error is 0.0024. The first sum has
two significant digits and the second sum has three significant digits.
1.7 1. 74.9967, 0.0133
2. -100.009, -0.0001
1.8 same as above
1.9
ln (1 + x)
Terms included x = 0.5 x=2
1 0.5 2
2 0.375 0
3 0.41666667 8/3
4 0.40104167 -4/3
Relative error in estimating ln (1 + x)
Terms included x = 0.5 x=2
1 0.2332 0.8295
2 -0.0751 -1
3 0.02763 1.4273
4 -0.01091 -2.2136
For x = 0.5, the error monotonically decreases be factor greater than 2.
For x = 2, the error is increasing with increasing number of terms. This method of estimating ln
(1 + x) is inefficient for x = 2, or actually for x > 1 because the value of the x term increases with
increasing power. This produces a divergent series. We cannot use this series to estimate the
function for x = 2.
1.10 Demonstrated for x = 0.5. Sum of the series for 3 terms is 0.4794 using the relative error
criterion. In situations where we do not know the true answer, we may use the stopping criterion
(last term)/(summation of terms) < 0.001. In that case, for x = 0.5, 4 terms are required to
produce a solution 0.4794 that meets the criterion for convergence. The tolerance specification
for the 2nd criterion is at least as stringent as the 1st.
1.11 f '( 2) = 33. Estimation of first-order derivative using
forward difference: 37.584
backward difference: 28.876
central difference: 33.23
method absolute error relative error
forward difference 4.584 0.139
, backward difference 4.124 0.125
central difference 0.23 0.007
1.12 The range is from 56720 to 67150 or 0.9068 to 1.074 times the average. Only 4 significant
figures can be retained.
1.13 CD 34+ cells
Rolling velocities
2.399
2.817
0.815
2.321
0.972
2.670
1.368
1.985
3.276
1.357
Max: 3.276 absolute deviation = 1.278 relative deviation = 0.6396
Avg. 1.998
Min: 0.815 absolute deviation = 1.183
CD 34 – cells
Rolling velocities
2.134
2.672
4.386
2.680
2.755
2.091
2.567
2.616
2.032
2.897
Max: 4.386 absolute deviation = 1.703 relative deviation = 0.6347
Avg. 2.683
Min: 2.032 absolute deviation = 0.651
The average rolling velocities of CD 34- cancer cells is higher than that of CD 34+ cells by
0.685. We would need to perform statistical tests to identify if the differences are by chance or
actually signify differences in the respective populations.
SOLUTIONS MANUAL
,Solutions Manual for Chapter 1
1.1 Program:
Defining Constants
fraction = 0.2; %The decimal fraction to be converted to binary
digits = 16; % Number of binary digits in the binary number to be determined
binary(1:digits)=0; %The binary significand
Determining the binary significand
for i = 1:digits
base2power = (1/2)^i;
if (fraction >= base2power)
fraction = fraction - base2power;
binary(i) = 1;
else
binary(i) = 0;
end
end
16-digit binary significand for 0.7 is 0.1011 0011 0011 0011
16-digit binary significand for 0.2 is 0.0011 0011 0011 0011
1.2 (i) (a) 0. 0 0 0 1.
(b) 0. 1 0 1 0
(ii) 0. 1 0 1 1
(iii) 0.6875
(iv) relative error: 11.61%
1.3 (a) 1.1001 × 2110
(b) 1.010000000010101100000010000011000100100110111010011 × 2110
(c) 1.0000 × 2-110.
1
1.4 relative error using 6 significant digits is 0.2 or 20%. Instead by evaluating ,
x + 1 + x2
2
we get a relative error of -0.000203 or -0.0203%. The rearrangement reduced the error by 1000
times or 3 orders of magnitude!
1.5 The first expression gives us 7.1 and the second expression gives us 7.106. The second
expression produces a more accurate result. The loss of accuracy is due to the multiplication
,steps, where we lose information due to 4-digit rounding arithmetic. The first expression
involves 7 multiplication steps. The 2nd expression involves only 2 multiplication steps.
1.6 Absolute error is 0.0076. After summing the smallest terms first, and in the order of smallest
to largest numbers, maintaining 3-digit rounding, the absolute error is 0.0024. The first sum has
two significant digits and the second sum has three significant digits.
1.7 1. 74.9967, 0.0133
2. -100.009, -0.0001
1.8 same as above
1.9
ln (1 + x)
Terms included x = 0.5 x=2
1 0.5 2
2 0.375 0
3 0.41666667 8/3
4 0.40104167 -4/3
Relative error in estimating ln (1 + x)
Terms included x = 0.5 x=2
1 0.2332 0.8295
2 -0.0751 -1
3 0.02763 1.4273
4 -0.01091 -2.2136
For x = 0.5, the error monotonically decreases be factor greater than 2.
For x = 2, the error is increasing with increasing number of terms. This method of estimating ln
(1 + x) is inefficient for x = 2, or actually for x > 1 because the value of the x term increases with
increasing power. This produces a divergent series. We cannot use this series to estimate the
function for x = 2.
1.10 Demonstrated for x = 0.5. Sum of the series for 3 terms is 0.4794 using the relative error
criterion. In situations where we do not know the true answer, we may use the stopping criterion
(last term)/(summation of terms) < 0.001. In that case, for x = 0.5, 4 terms are required to
produce a solution 0.4794 that meets the criterion for convergence. The tolerance specification
for the 2nd criterion is at least as stringent as the 1st.
1.11 f '( 2) = 33. Estimation of first-order derivative using
forward difference: 37.584
backward difference: 28.876
central difference: 33.23
method absolute error relative error
forward difference 4.584 0.139
, backward difference 4.124 0.125
central difference 0.23 0.007
1.12 The range is from 56720 to 67150 or 0.9068 to 1.074 times the average. Only 4 significant
figures can be retained.
1.13 CD 34+ cells
Rolling velocities
2.399
2.817
0.815
2.321
0.972
2.670
1.368
1.985
3.276
1.357
Max: 3.276 absolute deviation = 1.278 relative deviation = 0.6396
Avg. 1.998
Min: 0.815 absolute deviation = 1.183
CD 34 – cells
Rolling velocities
2.134
2.672
4.386
2.680
2.755
2.091
2.567
2.616
2.032
2.897
Max: 4.386 absolute deviation = 1.703 relative deviation = 0.6347
Avg. 2.683
Min: 2.032 absolute deviation = 0.651
The average rolling velocities of CD 34- cancer cells is higher than that of CD 34+ cells by
0.685. We would need to perform statistical tests to identify if the differences are by chance or
actually signify differences in the respective populations.
