exercises-42-44-show-how-to-use-the-condition-number-of-a-matrix-a-to-estimate-the-accuracy-of-a-c-2

Question

Exercises $42-44$ show how to use the condition number of a matrix $A$ to estimate the accuracy of a computed solution of $A \mathbf{x}=\mathbf{b}$ . If the entries of $A$ and $\mathbf{b}$ are accurate to about $r$ significant digits and if the condition number of $A$ is approximately $10^{k}$ (with $k$ a positive integer), then the computed solution of $A \mathbf{x}=\mathbf{b}$ should usually be accurate to at least $r-k$ significant digits.
[IM ] Repeat Exercise 42 for the matrix in Exercise 10 .

Ahmad Reda · Accepted Answer

in Problem 43. We have these metrics square metrics five by five We call it metrics A. We will take any random victor X where X equals B. Then X must be in are five. Then we will take X equals. For example one zero minus five eight minus two. Then the multiplication off A and X gives the victor B and after multiplication multiplying road by column Road by column It&#x27;s row by the vector. It&#x27;s true by the victor, for example, we have five plus zero minus five equals zero Lost seven, multiplied by eight equals 56 minus 18 equals 38 and so on to get 76. 54 buying a 73 78. Then we will use a metrics program. Have you used Excel to get X for example? Now, if we don&#x27;t know X, we don&#x27;t know this Victor. We will use the program to get to solve for X here and call it X one from Excel Way Get X one equals using the inverse off equals in verse about to blow it by B and it gives this ends These victim for 11. This milk places it gives one 000 until four in the 11. This small place and it gives minus a point or or or or until without six in the 11th, this small place. Then we have minus 4.999 until mine in the elefant, This small place. And we got eight point. Oh, and finally of minus two point or Oh, it approximately equals one minus zero, which is zero minus 4.5. It, um, minus two. This is accurate. It&#x27;s accurate. Toe nine, this woman places because if we take the Kirsty to 10 decimal places, this will give us here zeros until we got one. In the end, which is not accurate to zero, that it&#x27;s accurate toe mind this small places. And this is the final answer off our group.

Runpeng Li · Answer

this problem. We need to use me to use that waiting to program in this case, Um, it&#x27;s really hard to explain just on the paper. I&#x27;ll just trauma try my best to explain to you. So, um, here were given the Matrix. That is, um, from exercise nine. So it is or zero negative. Seven nick of seven and make it six 1 11 nine and seven. Sorry. Seven. Next. Five, 10. 19. The last row is selected. 1231 Okay, so the random vector that I used in this case, um, I&#x27;ll try to write it down because we the vector, the valley off the off the each entry I generated by the random function in in some pie I The thing that I got has 18 digits, so I just tried to, uh, ready down approximately. So the factor that I generated is to appoint to aid one and do a point by by one. And is there a point by zero? Bye. And excuse me, is there a point to some date? So this my random vector, Um, And for the more we find, find out a X equals B. So get B in my case. Um B, it&#x27;s ah. Again, I will try to, uh, right down the approximation here. That is next. Four point three, 56 and six point ny 21 and nine point by four to and two point zero. Fine. Eight. Okay, so this is my book approximation for accent now. Oh, I mean, to use the matrix to compute the solution or given a and b uh, are a It&#x27;s just a this matrix. The rock formations are B will be this factor here. Okay? And I saw for the system the X one that I got from right here. It&#x27;s almost the same. I&#x27;ll just again I&#x27;ll try Thio, right that write it down as a approximation. Um, so the 1st 3 the three digits behind behind the decimal point is the same. So, uh, this is how I all right down here to a point 281 and your point. Bye bye. One and your point by duel. Fine. And is there a point to some date? Okay, um, I cannot you know, I cannot write down the entire decimal, Um, just on this page. But what I can say from my result is that the difference off X one minus acts is something that let me check. Um, the difference is some dismal. That that is very close to zero. I&#x27;ll try to write down the prosecution. Here is negative too. Point one 93 times 10 to power off 14. And, um, second entry. We have 6.3 eight times 10 to the power. Oh, 13 and second entry. We have three point one one u times and to the power off. Negatives. 13 and the last country we have elected three point 2 37 times 10 to do power on 13. So you can see that. Um, since the condition number, I forgot to include the previous material, but a condition number. Condition number off A. What I got for condition number is 23 682 point ny two eight. So that kind of condition number is quite large is just a, um five digits, five digit number, and, uh, so that results in our difference or ex one acts is quite a small. It&#x27;s just to the power for team to borrow. 13. So it&#x27;s, uh, what I&#x27;m concerned about this solution is that x one X x one and x are close to each other and miracles in this case. So this is what what I got for our program So you can try your my own random better and try your own. Uh, here we you will have your your own Dr B. And you can see you Who You got the same similar similar solution to me because I got I got a decision that says X one and acts are pretty close to Java. You can compare our compared this bit of difference to my to my difference and see if that&#x27;s if that&#x27;s the same thing to you.

Runpeng Li · Answer

bring this problem again. We need to use our programming skills to solve the problem. Um uh, I&#x27;ll try my best to explain it. And the first thing we need to do is to find out, um, solved the question. They actually will be for a suitable be to find a nest column off of the Ebro itself. Beef outer, older Hubert Matrix. So they hear of me tricks. All right, down here. So the huber Beatriz we were considering in this problem is a five by five matrix 11 have 1/3. What? Her 1/5? Okay. Ah, Nde. Same for the column Here. 1/2 1 3rd and one 1/4. I don&#x27;t want it. Excuse me. And the second row would be 1/3. 14 one kid 16 and same for the column. Okay. What? Six. And the third row would be one too. 16 17 one Stay. 17 and 171 eh? One a And one night. Okay. So, um, for a suitable be we&#x27;re taking here. Seems we we&#x27;re considering too solve a system by on, uh, we&#x27;re using the super B to south a system and find out the last column lopped off the inverse of matrix A. So the way to do that, the suit both be we&#x27;re taking here. It&#x27;s all zeros. But in the country is one why we need to take it, Inspector. Well, because so consider the lunar system here, a axe equal Speak now to solve that we take the universe. So that is, um, members of a time speed. Um, so the inverse of age to be should be something some mysteries with the The Nats column here times Um, don&#x27;t just mark the last column as the as a vector say vector A and times the vector beat. So we called at the hour. Our vector bees, all zeros. But the centuries want so 00 and one. So, um, what will be the product off these two music&#x27;s well, the product will be Think about the previous thing. This should be because the Virgin The first column. We take the product over the first column for century and zero, the second entry and zero the third entrance. You know, So all of these previous things will be zero. So that&#x27;s zero procedural, but zero. We have four zeros here. Plus, until when we come to the last entry For the first roll, we have one. So that is exactly the last column that say on a fight. So similarly, we have, um Ah. Oh, sorry. Should be a one. Should be a one. Yeah. Do you want, um, similarity? We will have all the other on entries for a defector. And until the last thing is P five, So that is a one two, a five. And this is our vector Big victory. And our vector is is exactly the last column off a burns off. So, yeah, this is this is our idea, and bye bye. My computation. Here I find out, um, Sector A is something like this. 6 30 6 30 and connective. Um, 12,600 and 56,000 and 700 and negative. 888 8200. Okay. 1002 100. All right. The last country is 4000. 44,100 44,000. 100. Okay, so this should be the last column off the humerus off, eh? And so, for the next question, how many digits in each entry off acts do we expect to be correct. Um, the thing is, to the first thing is to we need to find a condition number off, eh? By my confusion, I found out condition. Number is it&#x27;s about 47 66 zero and seven. So, uh, an hour and in my condition, I is the is the approximation that it&#x27;s, uh uh, That keeps us 15 significant digits. So, uh, the accuracy I would I would expect is 15 minus. Here we have 123456 So 15 minus six. So I would expect the theocracy will be seven atleast 90 jizz each entry of X. So that&#x27;s ours.

James Chok · Answer

Okay, So in this question, you want to approximate our Ivan values and Ivan vectors off this a metrix right here. And by the way, So this is a This is really important. So if you want to use it, just use plastic. So this is a matrix right here on the day of that is given to us is zero is equal to 11 x zero x one is 10.736 and so on. So on, so on. And so what this step does here is we multiply a and zero x want eggs, Irving this, then you do X one x to x ray and sometimes awe. Then what this stuff does it is that it essentially returns the largest valley in here in absolute terms. Yeah, and so would print out the results. So first thing that would print out is the multiplication together, and we&#x27;ll print out in your second. So just be careful when I wanna print out that I printed out the transpose not be better. So you&#x27;ll get a one by two vector instead of two by two. Victor. The reason why because just by the sanitation is a lot easier to read, so what we&#x27;ll see is x zero is negative. 1.9 and a times by x zero is negative 1.9 and negative 1.4 large entreating next, 1.9 and so on. So on, so on and finally x for a times x four is negative. Zero point negative 0.0.4012 on negative 0.3008 with life&#x27;s entry being negative. 04012

Gennady Notowidigdo · Answer

Okay, so we are given This are four by four matrix, which I will hard light in green. So what, we&#x27;re gonna do it be going to use a computer package? So I used Mabel. If you want to use Matt, Matt or Mathematica, I feel free to do so or something else off that, like again, feel free to use whatever computer package you desire. So once I have this four by formation, so I want to find the Eiken values and the corresponding Aiken vectors, or sort of a basis for the Eiken space, so to speak. All right, so two. So, once I used the computer package, so let me just sort of figure out what this character is. It polynomial is. So if I obtained the characteristic polynomial, it&#x27;s going to be our to the four miles to are cute, minus 311 are square plus 312 aw, plus, 24,000 336. And I&#x27;m sure that this fact is nicely, um, so factors into, uh, plus 12 times our minds 13. A love that square good. And when we solved the zeros off the characteristic polynomial which is in this case minus 12 and 13 are then what we called the Aiken Valleys. So for this particular matrix, we have to Aiken values minus 12 and 13 case are so that our oppressed it yet. Okay, So for the Eiken battle, it&#x27;s for the Eiken value. Say Lambda equals two. I&#x27;m with these arms. That lab, just to make it easier, are equal to 13. Then if we compute, uh, our time, the identity matrix minus the matrix that we have, let&#x27;s call it a Okay, so let&#x27;s call it a Are I minus A. This ends up being for 4 to 4. 56 minus 19 28 minus 44. And then we just raised this just to make make it need us to be minus 19. Um, just dish it black. It&#x27;s minus 19. Then you got 14. 14 seven on 14. That should be 14 uh, minus 42. 33 Mona&#x27;s 21 and 58. Okay, now the Eiken space for our the Eigen value, our eagles 13 is pretty much just the null space off our times. I minus a or in other words, the Colonel. Okay, if you want you can. You can interject two things. Although I reserve the word colonel for lenient maps and no spaces for matrices. Okay, Now this matrix reduces too to to want. So I&#x27;ve done it in its simplest form. So it ends up reducing to something like this. Okay, so at this point, you can use back substitution and you&#x27;ll end up for the basis for the null space ends up being It will be minus 10 to 0 and one minus 40 and three. So obviously, if you did back substitution, you would set these two quantities to be one, and you work backwards, but I&#x27;ve just sort of normalized. It&#x27;s another normal, but I&#x27;ve made it fraction free just for our simplicity purposes. Okay, so we&#x27;ve got a basis for the eye Gance base for articles 13 and similarly, I will do in the next page, sir. I&#x27;ll keep it in black financing for the Eigen value. Let our used ob for I mean, it doesn&#x27;t really matter. I just like using are because we&#x27;re sticking to English letters rather than moving to Greek letters. But in any case, so for the Eiken value are equal to minus 12. Are I modest, eh? You know, when you write out what its people to sort of figure that out on your own. But it reduces to Mona&#x27;s 21 for 24 Sara Wan. Modest one of one. And in two bars, off zeros on a basis for the null space. So for the Eigen space for article 2 12 minus 12. Sorry is 2770 and zero. Modest one, sirrah. One. Okay, so I&#x27;ll leave it there. Uh, please verify older cat places that I&#x27;ve made Adlai Vilo on the computer or by hand, But in any case, thank you very much.

Jack Chen · Answer

Okay, So for this problem, we just need to put our Matrix into the calculator. And the calculator real directly gives us this result. And as we can see, the single values off on this metric say will be on this for numbers and the corresponding single ability composition will be some chocolates.

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Exercises $42-44$ show how to use the condition n…

Problem 43 Easy Difficulty

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

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Maria G.

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Absolute Value - Example 1

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Linear Algebra and Its Applications

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Absolute Value - Example 1

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David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Catherine R.

Heather Z.

Maria G.

Kristen K.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications