Lower Colorado River (Problem 11.12 in Chapra) My Solutia The Lower Colorado River consists of a series of four reservoirs as shown in Fig. P11.12 in the textbook. Mass balances can be written for each reservoir, and the following set of simultaneous linear algebraic equations results: [ 13.422 0 00; -13.422 12.252 0 0; 0-12.252 12.377 0; 00-12.377 11.797] * [C_1, c_2, c_3, c_4)' = [750.5, 300, 102, 30) where the right-hand-side vector consists of the loadings of chloride to each of the four lakes. The variables c_1, c_2, c_3, and c_4 are the resulting chloride concentrations for Lakes Powell, Mead, Mohave, and Havasu, respectively. Given this information, complete the following tasks: (a) Use the matrix inverse to solve for the concentrations in each of the four lakes. (b) How much must the loading to Lake Powell be reduced for the chloride concentration of Lake Havasu to be 75? (c) Using the matrix 2-norm (spectral norm), compute the condition number and how many suspect digits would be generated by solving this system. Do not round number of significant digits for part (c). You may use the MATLAB condition number function (cond). Hint for part (b): Do not use a negative sign to indicate a reduction, a positive number is expected. Hint for part (c): In MATLAB log is the natural log function, whereas log10 is the base 10 log function.

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Lower Colorado River (Problem 11.12 in Chapra)
The Lower Colorado River consists of a series of four reservoirs as shown in Fig. P11.12 in the textbook. Mass balances can be written for each reservoir, and the
following set of simultaneous linear algebraic equations results:
[ 13.422 0 00;
-13.422 12.252 0 0;
0 -12.252 12.377 0;
00-12.377 11.797]
* [C_1, c_2, c_3, c_4]'
= [750.5, 300, 102, 30]
where the right-hand-side vector consists of the loadings of chloride to each of the four lakes. The variables c_1, c_2, c_3, and c_4 are the resulting chloride
concentrations for Lakes Powell, Mead, Mohave, and Havasu, respectively.
Given this information, complete the following tasks:
(a) Use the matrix inverse to solve for the concentrations in each of the four lakes.
(b) How much must the loading to Lake Powell be reduced for the chloride concentration of Lake Havasu to be 75?
(c) Using the matrix 2-norm (spectral norm), compute the condition number and how many suspect digits would be generated by solving this system.
Do not round number of significant digits for part (c). You may use the MATLAB condition number function (cond).
Hint for part (b): Do not use a negative sign to indicate a reduction, a positive number is expected.
Hint for part (c): In MATLAB log is the natural log function, whereas log10 is the base 10 log function.
Transcribed Image Text:My Solutions > Lower Colorado River (Problem 11.12 in Chapra) The Lower Colorado River consists of a series of four reservoirs as shown in Fig. P11.12 in the textbook. Mass balances can be written for each reservoir, and the following set of simultaneous linear algebraic equations results: [ 13.422 0 00; -13.422 12.252 0 0; 0 -12.252 12.377 0; 00-12.377 11.797] * [C_1, c_2, c_3, c_4]' = [750.5, 300, 102, 30] where the right-hand-side vector consists of the loadings of chloride to each of the four lakes. The variables c_1, c_2, c_3, and c_4 are the resulting chloride concentrations for Lakes Powell, Mead, Mohave, and Havasu, respectively. Given this information, complete the following tasks: (a) Use the matrix inverse to solve for the concentrations in each of the four lakes. (b) How much must the loading to Lake Powell be reduced for the chloride concentration of Lake Havasu to be 75? (c) Using the matrix 2-norm (spectral norm), compute the condition number and how many suspect digits would be generated by solving this system. Do not round number of significant digits for part (c). You may use the MATLAB condition number function (cond). Hint for part (b): Do not use a negative sign to indicate a reduction, a positive number is expected. Hint for part (c): In MATLAB log is the natural log function, whereas log10 is the base 10 log function.
1 function [A_inverse,c, del_b1, cond_num, digits]=Chapra_11p12()
2 %% Output
3 % A_inverse: 4x4 Inverse of coefficient matrix
4 %
C:
4x1 solution vector [c1; c2; c3; c4]
5 % del_b1:
6 % cond_num:
7 % digits:
reduction in Lake Powell loading (scalar)
condition number of A, based on spectral norm (scalar)
number of suspect digits in solution, c (scalar)
9 % HINTS: Do not use a negative sign to indicate a reduction for part (b)
10 %
Do not round number of significant digits for part (c)
11 %
Note that, in MATLAB, log is the natural log function,
12 %
whereas log10 is the base 10 log function.
13
14 % Write your code here
15 A=[13.442 0 0 0;
-13.442 12.252 0 0;
O -12.252 12.377 0;
0 0 -12.377 11.797];
16
17
18
19 W=[750.5,3e0,102, 30];
20 C- AI*W
21 end
22 end;
23
Transcribed Image Text:1 function [A_inverse,c, del_b1, cond_num, digits]=Chapra_11p12() 2 %% Output 3 % A_inverse: 4x4 Inverse of coefficient matrix 4 % C: 4x1 solution vector [c1; c2; c3; c4] 5 % del_b1: 6 % cond_num: 7 % digits: reduction in Lake Powell loading (scalar) condition number of A, based on spectral norm (scalar) number of suspect digits in solution, c (scalar) 9 % HINTS: Do not use a negative sign to indicate a reduction for part (b) 10 % Do not round number of significant digits for part (c) 11 % Note that, in MATLAB, log is the natural log function, 12 % whereas log10 is the base 10 log function. 13 14 % Write your code here 15 A=[13.442 0 0 0; -13.442 12.252 0 0; O -12.252 12.377 0; 0 0 -12.377 11.797]; 16 17 18 19 W=[750.5,3e0,102, 30]; 20 C- AI*W 21 end 22 end; 23
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