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comparison of false position and secant methods

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PG1995

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Hi

Could you please help me **broken link removed** queries?

In case of the Q1, my personal guess would be that the mathematical curiosity was the primary force behind the invention of these numerical methods. Thank you.

References:
1: https://en.wikipedia.org/wiki/Secant_method
2: https://en.wikipedia.org/wiki/False_position_method
3: https://www.electro-tech-online.com/custompdfs/2012/10/lecture6-1.pdf

Regards
PG

PS: Please note this is not a question or related to any of the queries above; it is for only general reference. If you are confused by what the Wikipedia article says here about the false position method, then this PDF could be of some help. Thanks.
 
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Q1. The methods are even more useful in an age without computers because a person can carry out the calculations by hand. In those days a good method converged quickly and saved the person a lot of sweat, tears and time. Nowadays we don't generally use or even learn all of these methods. Personally, I might use Newton's method, or I'll write a program that scans the range for sign changes and then does a binary search between the sign changes. This is very simple to program and easy to understand, and converges very quickly.

Q2. I didn't try to check if you explained the methods correctly. What you said seems logical, but I dont' want to spend a lot of time to check something both you and I will never use. Hopefully, someone else is more familiar and will check them for you.

I agree that the RF method has these good attributes you mentioned. Perhaps the other method is used if a person is lazy or if the problem does allow someone to easily find boundaries where there are sign changes. Again, in today's world this is not too important. We can easily plot graphs, and let computers do the hard work for us.
 
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... in today's world this is not too important. We can easily plot graphs, and let computers do the hard work for us.

As i reread this part, I don't like the way I made it sound. By this statement, I don't mean it is not important to be familiar with the various methods and to understand the basic principles by which they work. There is great value in seeing how creative people found creative solutions in the past. These examples serve as inspiration for us to find creative solutions to our own problems . Also, when you study what they did, you are storing away methods in the back of your mind, and that might help you solve your own problems in the future.

What I do mean to say is that (in my opinion) we don't need to study these methods in great detail and become masters of them. You are very unlikely to actually use them in your life, but very occasionally a particular problem may lend itself to one of these "less-popular" methods. If you have at least seen and understood them in the past, your mind will be triggered to remember them at the critical moment and then you can go back and learn the details as needed.

Having said all this, if you are required to learn them in detail for your class, then by all means learn them well and Ace your test.
 
Hi

Could you please help me with **broken link removed** query about the secant method? Please note that I'm only concerned about the designation of the points at the start of the method. Thanks a lot.

Best wishes
PG
 
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I think that answering this question requires understanding the derivation of the method and understanding the requirements of the two initial guesses. My understanding of the secant method is that it is derived from the Newton-Raphson method, but the two guesses are required to approximate the derivative needed. This method is preferred when symbolic determination of the derivative of the function is not practical for some reason. Without verifying carefully, I think the method does not restrict the two guesses to have any requirements, such as bounding the root or requiring one to be left or right of the root.
 
algorithm for secant method

Hi

Could you please help me with **broken link removed** query? Thank you.

Regards
PG
 
Seems to me that it should be started with i=1 also, but it's a minor issue. I'd guess the program developer decided that the second guess is the first iteration and prefers that the index i refers to next estimate.
 
Hi

There are mathematical formulae to see if a certain method is converging or not. But let's look at it informally. Can successive iterations help us to see if a method is convergent or not for some problem? Suppose, we are finding a root of a function f(x). At iteration #13 we have f(x_13)=0.04, and at iteration #14 we have f(x_14)=0.5 and at iteration #15 f(x_15)=0.6, but at iteration #19 the function f(x_19)=0.0001. The iterations #14 and #15 might suggest that the method has started diverging because the result at the iteration #13 was closer to the root, and one may stop going on with more iterations concluding that the method is not applicable. But I think this is not true. I have seen sometimes at some successive iterations the results indicate as if the method has started diverging but then all of a sudden there is a change and convergence is apparent. So, in a nutshell, to see if a method is convergent or not, one has to look at a set of iterations, say 20, 30, 100, etc., rather than just two or three iterations. For example, if over a range of 50 iterations there hasn't been any refinement or there is more divergence then the method is divergent. Is my thinking correct? Please let me know. Thank you.

Regards
PG
 
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Hi

There are mathematical formulae to see if a certain method is converging or not. But let's look at it informally. Can successive iterations help us to see if a method is convergent or not for some problem? Suppose, we are finding a root of a function f(x). At iteration #13 we have f(x_13)=0.04, and at iteration #14 we have f(x_14)=0.5 and at iteration #15 f(x_15)=0.6, but at iteration #19 the function f(x_19)=0.0001. The iterations #14 and #15 might suggest that the method has started diverging because the result at the iteration #13 was closer to the root, and one may stop going on with more iterations concluding that the method is not applicable. But I think this is not true. I have seen sometimes at some successive iterations the results indicate as if the method has started diverging but then all of a sudden there is a change and convergence is apparent. So, in a nutshell, to see if a method is convergent or not, one has to look at a set of iterations, say 20, 30, 100, etc., rather just two or three iterations. For example, if over a range of 50 iterations there hasn't been any refinement or there is more divergence then the method is divergent. Is my thinking correct? Please let me know. Thank you.

Regards
PG

What you are saying is generally the case, but in mathematics you can't make proofs in this way. As you say, methods are informal when you don't quantify convergence. I think that engineers and scientists are much more willing to use such methods compared to mathematicians, as they would demand that errors be quantified and bounded to prove convergence. Perhaps this is because, when you are doing something practical, you generally have good guidelines on how accurate you need to be and whether the answer you get is accurate enough.
 
jacobi method for solving system of linear equations

Hi

Could you please help me with this query? Thank you.

Regards
PG
 

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I think it is supposed to be j=1, not j-1.

Thanks a lot, Steve.

I believe you have it right. But don't you think it would have been more correct and clear if written as below? Thank you.

jacob7-jpg.70269


Regards
PG
 

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The three ways I've seen are as follows.

[latex] \sum_{j=1,j\ne i}^n[/latex]

[latex] \sum_{j,j\ne i}[/latex]

[latex] \sum_{j\ne i}[/latex]

In other words, if you place the upper limit "n" on top, you should place the lower limit on the bottom along with the index. Then any restrictions (such as j\=i) can be placed at the bottom. Usually, the qualifier j\=i is bellow the j=1, but I didn't want to figure out how to do that in latex.

The last method only shows the terms that are not summed and it is implied that all other allowed values of j, are summed. BTW, I prefer the last method, since the context almost always implies all values of j, and the range of j is generally clear.
 
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