Question

Problem 3. Matrix condition The Lower Colorado River consists of a series of four reservoirs as shown in Fig. P11.12. Mass balances can be written for each reservoir, and the following set of MENG 602/438 Hao-Chun Chiang simultaneous linear algebraic equations results: Ac=b [ 13.422 0 0 0 -13.422 12.252 0 0 0 - 12.252 12.377 0 0 -12.377 11.797 (750.5 300 102 30 where the right-hand-side vector consists of the loadings of chloride to each of the four lakes and c1, c2, c3, and c4 = the resulting chloride concentrations for Lakes Powell, Mead, Mohave, and Havasu, respectively. (0) Use the matrix inverse to solve for the concentrations in each of the four lakes. Find Euclidean norm of A and A (iii) Calculate matrix condition number using below equation, and how many suspect digits would be generated by solving this system. Cond[A] = || A||- ||- (Optional) Compute the condition number using Frobenius norm definition. Compare to the result solved by full matrix and rounded matrix.

Answer 1

- Answer 13.422 0 2 o 13.422 0 -13.402 692.052 0 7 0 1 cl 1-13.422 12.252 300 o 102 0 - 12. 377 Lo -2379 1.797) lo] 11.797 30 13.422 xc, = 750.5 - -13.402 x C, + 12.252x (2 = 300 - -12.252 x C + 10.377x Cz = 102 - - 12.377* (₂ + 11.797x C 4 = 30 - ④ 0=) < = 750.5 13.4222 55.9 C,

D=) -13.५२२x 55:१ + 12. २5२ x C, = 300 10.25 २ x ८, = 300 + 250.२१ C, - 1050.29 । २. 25२. C, = 85.१२ . 3 ) - 1२ . २5२ x 85.70 + 10. 379 x C, = 10 २ । २. 377 x ८९ - 102 + 10 50. २५ ___C3 - 115 2. 24 2. 377 C3 - 93.09 @ =) - 12.3२१ x १3.0१ + ।। ११२ x C ।। 297 x C = 30 = 30 + 115 2. 17

Cy = 1182.17 11.797 = 100.2 y Fotter i) Powell = 55.9 = C, Mead = 85.72 = 2 Mohave = 93.09 -. Cza Havasu's 100.2 = CH i Å ' . 70.0745 0 0 0.0816 : 0:08160 0.0808 0.08 og 0.0808 10.0848 0.0848 0.0848 0.0848/

Euclidean norms of A = UJALI - (3.4, 5 13 4.2.) เอ.54 , (-2.2535 13.357 6 12 22 ) 4 11 12 ) - 10 6 0 3 = 33. 36 Evelideen norm of À DUAL - 06-0945 + 0.06 1 0.608 1 6. 0816 + 6. %0$ * 0 + ซ. 0 %0 % 0 0 $ 6 6. 0 $+ O . 0949 + 6: 0 % ฯ 1 0.051 6. 08 ) =J6:06 3 = 9.253 - ri) Cord [A] = \LA 11-11 4 11 33.46 x 0.99 = %. 6