Tridiagonal matrix Puns

How do I put the following matrix into a tridiagonal one using partial pivoting and row operations

First I swapped rows 1 and 3 to get the largest absolute pivot/diagonal entry

Then I took R2 <-- 2R2 + R1

I'm unsure how to get that 21 in the bottom position in the last column as it will effect the diagonal entry in column 2. Unless I am doing this wrong, any help is appreciated.

I am hoping someone can help me begin to address my deficiencies in understanding this problem. On Math Stack Exchange, the OP is asking how to find eigenvalues and eigenvectors of block Toeplitz matrices. I have a similar problem I wish to solve, but I would like to understand their work first, so my question is this:

How did the OP go from, T, a 2N-by-2N matrix, represented as the sum of tensor products (the Pauli matrix step) to T as the sum of (2-by-2) matrices after plugging in the eigenvalues of each (N-by-N) sub-block of T?

I understand that the eigenvalues of the tridiagonal Toeplitz matrices fill each of the elements of the 2-by-2 representation of T, and also that the eigenvectors of tridiagonal N-by-N Toeplitz matrices (e.g., A, B, etc) are all the same, and thus simultaneously diagonalize all blocks of T(2N-by-2N).

Thanks for the help!

I have a working subroutine for solving a tridiagonal matrix with periodic boundary conditions (this is the problem formulation). I have modified this subroutine in order to preserve the matrix. Here is what I have,

subroutine triper_vec(dl, dm, du, b, x, n)
    integer, intent(in) :: n
    double precision, intent(in) :: dl(:)   ! lower-diagonal
    double precision, intent(in) :: dm(:)   ! main-diagonal
    double precision, intent(in) :: du(:)   ! upper-diagonal
    double precision, intent(in) :: b(:)    ! b vector
    double precision, intent(inout) :: x(:) ! output

    double precision, dimension(n) :: w     ! work array
    double precision, dimension(n) :: maind ! used to preserve matrix
    integer :: i, ii
    double precision :: fac

    w(1) = -dl(1)
    maind(1) = dm(1)
    x(1) = b(1)
    do i = 2, n - 1, 1
        ii = i - 1
        fac = dl(i) / maind(ii)
        maind(i) = dm(i) - (fac * du(ii))
        x(i) = b(i) - (fac * x(ii))
        w(i) = -fac * w(ii)
    end do
    x(n) = b(n)
    maind(n) = dm(n)

    ii = n - 1
    x(ii) = x(ii) / maind(ii)
    w(ii) = (w(ii) - du(ii)) / maind(ii)

    do i = n - 2, 1, -1
        ii = i + 1
        x(i) = (x(i) - du(i) * x(ii)) / maind(i)
        w(i) = (w(i) - du(i) * w(ii)) / maind(i)
    end do

    i = n
    ii = n - 1
    fac = maind(i) + (du(i) * w(1)) + (dl(i) * w(ii))
    x(i) = (x(i) - ((du(i) * x(1)) + (dl(i) * x(ii)))) / fac

    fac = x(n)
    do i = 1, n - 1, 1
        x(i) = x(i) + (w(i) * fac)
    end do

end subroutine triper_vec

Are there any glaring issues that could lead to performance increases? Or is there anything I can do to allow the compiler to produce a more optimized result? I am compiling with

gfortran -march=native -O3 triper.f90

Hello,

I am having trouble showing that the eigenvalues of a tridiagonal matrix are all strictly positive. The tridiagonal matrix has 2's along the main diagonal and -1's directly above and below the main diagonal, and every other entry is 0. Help is appreciated!

Hi I am trying to generate an arbitrary Gauss quadrature rule by using the Golub-Welsh algorithm (here). I need to code this on C++ for my personal project. This algorithm involves the eigenvalue decomposition of a matrix in which the only non-zero elements are the subdiagonal and superdiagonal. To illustrate in Matlab code: n = 16; beta = .5./sqrt(1-(2*(1:n-1)).-2); T = diag(beta,1) + diag(beta,-1); [V,D] = eig(T); I want to implement the eigenvalue decomposition in code and not use Matlab routines for this since I want to parallelize it. What is the best way to do eigenvalue decomposition for this type of matrix? Is bisection method acceptable for my use case? How about divide and conquer or QR method or Lanczos? I expect the n to be upto 512.

EDIT: the MRRR technique was pointed to me in another forum. It is a relatively new technique developed by Dhillon and Parlett. Apparently it has extremely good time complexity and has a guaranteed accuracy.

I can only seem to find subroutines for general or symmetric tridiagonal matrices. Is it just me? Any answers are appreciated.

Do your worst!

It really does, I swear!

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

Hi,

I have been experimenting a bit with an explicit and implicit Euler's methods to solve a simple heat transfer partial differential equation:

∂T/∂t = alpha * (∂^2T/∂x^2)

T = temperature, x = axial dimension. The initial condition I used is for x = 0, T = 100 °C. And the boundary condition at the end of the computational grid: for T**(x = L) =** T**(x=*****L-***1), where L is a length of the computational grid -> the last and the and next-to-last values of temperature are the same.

Would some be willing to look at my code (I am not a MATLAB guy, but I try to learn) whether my implementation of implicit method is correct.

My thoughts:

Explicit method (works fine): Every values of T are calculated by T1(i) + heat_coefficient*((T1(i+1)-2*T1(i)+T1(i-1))/dx^2)*dt, except for the first and the last value which are specified by the I.C. and B.C., respectively.

My question:

Implicit method: At first, I create a tri-diagonal matrix MAT
, which defines a relationship between the values in the next time line (n+1). In the case of an implicit method, I cannot write I.C. and B.C. exactly, and therefore I save them on the “Right side of the equation”, i.e. pS(i) = -(T2(i)*dx^2)/(heat_coefficient*dt);

By this, I express all the values in one time line (n) and after that, I continue in the following time line (n+1).

If I put I.C. or B.C. into the “Right side of the equation”, the results become very sensitive to any change of positional (dx) and time (dt) step. Behavior of the temperature function is therefore wrong: the curves of a temperature should converge to the same values after certain time (if the time goes to infinity), however, my curves converge to a different values (and randomly change based on dx and dt).

How do I achieve to start at a temperature of 100 °C, such that after an “infinitely” long time, all of the temperature curves converge to 100 °C?

Should the values of I.C. and B.C. be put into a tri-diagonal matrix - e.g., if I have matrix with dimensions [N,N], then I specify I.C. for point [1,1], and B.C. for point [N,N]? (Unfortunately, it did not work properly when I tried)

Also, my implementation of the tri-diagonal matrix is probably not very nice, but it works.

They’re on standbi

Buenosdillas

I'm surprised it hasn't decade.

Our code is given down below. We have been struggling wrapping our heads around how to write this matlab code. We cannot get the correct values and have tried multiple ways to find the correct values for u. Any help would be greatly appreciated :)

clear;

A = [3 5 0 0

1 4 6 0

0 5 7 3

0 0 3 8];

k = [13 27 43 41];

a = [];

b = [];

c = [];

n = length(A);

m = n-1;

for i = 1:n

f = A(i,i);

a(i) = f;

end

for i = 1:m

p = 1 + i;

s = A(i,p);

b(i) = s;

end

for i = 1:m

p = 1 + i;

s = A(p,i);

c(i) = s;

end

for i = 1:m

f = c(i)./a(i);

R = c(i) - f*a(i);

c(i) = 0;

a(i+1) = R;

end

u = [];

%u is supposed to equal [1 2 3 4]

https://preview.redd.it/genwag3ckgq61.png?width=621&format=png&auto=webp&s=759de8e747b8fec9c0183a2ce647849e35917dff

https://preview.redd.it/kqel5i3ckgq61.png?width=590&format=png&auto=webp&s=8bc9630ec4133650059996aea2de469bb5a5fe13

Pilot on me!!

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This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

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When I got home, they were still there.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

I won't be doing that today!

You take away their little brooms

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

There hasn't been a post all year!

Why

It’s pronounced “Noel.”

[Removed]

After all his first name is No-vac

What, then, is Chinese rap?

Edit:

Notable mentions from the comments:

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