Approximate Solution to a System of Linear Equations

alibaig1991 · March 12, 2018, 05:19

Hello everyone,

I hope you having a good day.

Consider a linear system, Ax=b where A is coefficient matrix, x is vector of variables whose values we want to find out, and b is a vector of constants in which all entries are same. Is there a way to find out relative values of xi (entries of x) without actually solving the system of equations. In other words, can say by just looking at A that x1 will be greater than x2 because the sum of entries of first row is greater than the sum of entries of 2nd row of A. I have intentionally made the assumption that all entries of b are same, so that they have no effect on our approximate solution.

Thank you!

FMDenaro · March 12, 2018, 05:31

Quote:

Originally Posted by alibaig1991

Hello everyone,

I hope you having a good day.

Consider a linear system, Ax=b where A is coefficient matrix, x is vector of variables whose values we want to find out, and b is a vector of constants in which all entries are same. Is there a way to find out relative values of xi (entries of x) without actually solving the system of equations. In other words, can say by just looking at A that x1 will be greater than x2 because the sum of entries of first row is greater than the sum of entries of 2nd row of A. I have intentionally made the assumption that all entries of b are same, so that they have no effect on our approximate solution.

Thank you!

The traditional way to get an approximate solution is to use iterative method wherein you do not invert the matrix. You can control the residual at each iteration.
I dont understand why are you considering the sum of the entries of A.

alibaig1991 · March 12, 2018, 06:34

Dear Filippo Maria Denaro,

I am trying to develop a stabilized method for hyperbolic equations using meshless methods. The method I described above is used in Finite Element Method and is known as Row Sum Lumping. I have tried to solve a system of equation using this method and the results are very inaccurate. I might have made a mistake. I will get back to you soon after revising the method.

Thank you.

FMDenaro · March 12, 2018, 06:39

Consider the vector of the residual at a certain k-th iteration. Each entry of this vector is the sum of the entries in the corresponding row of A, multiplied the components of the approximate solution at the stage k minus the components of the known term. In vector notation:

A.x_k-q= r_k

I suppose this is what you are trying to control, isn't that?

March 12, 2018, 05:19	Approximate Solution to a System of Linear Equations	#1
alibaig1991 New Member abcd efgh ijkl Join Date: Oct 2015 Posts: 26 Rep Power: 11	Hello everyone, I hope you having a good day. Consider a linear system, Ax=b where A is coefficient matrix, x is vector of variables whose values we want to find out, and b is a vector of constants in which all entries are same. Is there a way to find out relative values of xi (entries of x) without actually solving the system of equations. In other words, can say by just looking at A that x1 will be greater than x2 because the sum of entries of first row is greater than the sum of entries of 2nd row of A. I have intentionally made the assumption that all entries of b are same, so that they have no effect on our approximate solution. Thank you!

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March 12, 2018, 06:34		#3
alibaig1991 New Member abcd efgh ijkl Join Date: Oct 2015 Posts: 26 Rep Power: 11	Dear Filippo Maria Denaro, I am trying to develop a stabilized method for hyperbolic equations using meshless methods. The method I described above is used in Finite Element Method and is known as Row Sum Lumping. I have tried to solve a system of equation using this method and the results are very inaccurate. I might have made a mistake. I will get back to you soon after revising the method. Thank you.

March 12, 2018, 06:39		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,897 Rep Power: 73	Consider the vector of the residual at a certain k-th iteration. Each entry of this vector is the sum of the entries in the corresponding row of A, multiplied the components of the approximate solution at the stage k minus the components of the known term. In vector notation: A.x_k-q= r_k I suppose this is what you are trying to control, isn't that?