Developing the determinant on the first column, then the second minor on its first column, you obtain the key recurrence. Once you get a recursion formula there are at least three different ways to solve the problem. Using the Z transform, using Chebyshev polynomials, and using the traditional which I include here method to solve difference equations. This document illustrate this. From it, I will rewrite here one of the methods. We expand the determinant through the first row to find. However, and to simplify operations we can state simpler initial conditions.
Using the recursion 1. We apply the theory about difference equations to obtain the solution of our problem. We consider two cases;. That is, the quadratic equation is a perfect square. The recursive equation 1 has a solution of the form. One solution of this problem was given by Yueh  where all parameters are complex numbers. It does not uses polynomials, but instead the ring of formal sequences. Actually, Yueh's problem is more general because he finds the eigenvalues and eigenvectors of the matrix.
Once one has everything in terms of sequences, one can use the sum of sequence component-wise and the multiplication of sequences see below to find the solution of the problem. You can see this product also as one of the terms in the Cauchy product rule for polynomials. The explicit solution is obtained in a straightforward way by just doing algebra, see .
This is assured by Theorem 24 p. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Start collaborating and sharing organizational knowledge. Create a free Team Why Teams? Learn more. How to find the eigenvalues of tridiagonal Toeplitz matrix? Ask Question.
Asked 7 years, 8 months ago. Modified 9 months ago. Viewed 12k times. Rodrigo de Azevedo You should get enough information from Google. I search it but I can't find a good proof. For example, the following matrix is tridiagonal:. The determinant of a tridiagonal matrix is given by the continuant of its elements.
An orthogonal transformation of a symmetric or Hermitian matrix to tridiagonal form can be done with the Lanczos algorithm. A tridiagonal matrix is a matrix that is both upper and lower Hessenberg matrix. Although a general tridiagonal matrix is not necessarily symmetric or Hermitian , many of those that arise when solving linear algebra problems have one of these properties.
Hence, its eigenvalues are real. Many linear algebra algorithms require significantly less computational effort when applied to diagonal matrices, and this improvement often carries over to tridiagonal matrices as well. The determinant of a tridiagonal matrix A of order n can be computed from a three-term recurrence relation.
The sequence f i is called the continuant and satisfies the recurrence relation. The cost of computing the determinant of a tridiagonal matrix using this formula is linear in n , while the cost is cubic for a general matrix. The inverse of a non-singular tridiagonal matrix T. Closed form solutions can be computed for special cases such as symmetric matrices with all diagonal and off-diagonal elements equal  or Toeplitz matrices  and for the general case as well.
In general, the inverse of a tridiagonal matrix is a semiseparable matrix and vice versa. When a tridiagonal matrix is also Toeplitz , there is a simple closed-form solution for its eigenvalues, namely:  . A real symmetric tridiagonal matrix has real eigenvalues, and all the eigenvalues are distinct simple if all off-diagonal elements are nonzero. As a side note, an unreduced symmetric tridiagonal matrix is a matrix containing non-zero off-diagonal elements of the tridiagonal, where the eigenvalues are distinct while the eigenvectors are unique up to a scale factor and are mutually orthogonal.
For unsymmetric tridiagonal matrices one can compute the eigendecomposition using a similarity transformation. A transformation that reduces a general matrix to Hessenberg form will reduce a Hermitian matrix to tridiagonal form. So, many eigenvalue algorithms , when applied to a Hermitian matrix, reduce the input Hermitian matrix to symmetric real tridiagonal form as a first step. A tridiagonal matrix can also be stored more efficiently than a general matrix by using a special storage scheme.
From Wikipedia, the free encyclopedia. Matrix with nonzero elements on the main diagonal and the diagonals above and below it. Main article: continuant mathematics. Main article: tridiagonal matrix algorithm. A treatise on the theory of determinants. Dover Publications.
Once the decomposition is computed, you can use the matrixQ and matrixT functions to retrieve the matrices Q and T in the decomposition. Definition at line 61 of file Tridiagonalization. Definition at line 80 of file Tridiagonalization. Definition at line 89 of file Tridiagonalization. Definition at line 81 of file Tridiagonalization. Return type of matrixQ. Definition at line 99 of file Tridiagonalization.
Definition at line 70 of file Tridiagonalization. Definition at line 84 of file Tridiagonalization. Definition at line 66 of file Tridiagonalization. Definition at line 83 of file Tridiagonalization. Definition at line 69 of file Tridiagonalization. Definition at line 68 of file Tridiagonalization. Definition at line 96 of file Tridiagonalization. Definition at line 82 of file Tridiagonalization. Definition at line 72 of file Tridiagonalization. The default constructor is useful in cases in which the user intends to perform decompositions via compute.
The size parameter is only used as a hint. It is not an error to give a wrong size , but it may impair performance. Definition at line of file Tridiagonalization. This constructor calls compute to compute the tridiagonal decomposition. The tridiagonal decomposition is computed by bringing the columns of the matrix successively in the required form using Householder reflections. The cost is flops, where denotes the size of the given matrix. This method reuses of the allocated data in the Tridiagonalization object, if the size of the matrix does not change.
The Householder coefficients allow the reconstruction of the matrix in the tridiagonal decomposition from the packed data. This function returns a light-weight object of template class HouseholderSequence. You can either apply it directly to a matrix or you can convert it to a matrix of type MatrixType. Stack Overflow for Teams — Start collaborating and sharing organizational knowledge.
Create a free Team Why Teams? Learn more. Asked 6 years, 2 months ago. Modified 6 years, 2 months ago. Viewed 5k times. Add a comment. Sorted by: Reset to default. Highest score default Date modified newest first Date created oldest first. Chiel Chiel 5, 2 2 gold badges 29 29 silver badges 50 50 bronze badges. Of course it's true if I implement the algorithm myself it would be much quicker : I know this might sound kinda bad, but right now though I'm only interested in something that's quick to implement into the code which can also compute the solutions of the system faster than what I currently have, only because I have several other codes which I have to write up in very little time In that case, go for the BLAS?
GTSV routine. On second thought the Thomas algorithm does look quite easy to implement, I didn't read through it properly at first but it actually looks extremely easy and quick to write up. Thanks for telling me about it! ThePlowKing it is really only a few lines of codes and the algorithm is super fast. You can find a vectorized version that I have written myself in the following file function name tdma : github. Thanks heaps for that! Show 3 more comments.
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