Gershgorin and Rayleigh Bounds on the Eigenvalues of the Finite-Element Global Matrices via Optimal Similarity Transformations

The large finite element global stiffness matrix is an algebraic, discreet, even-order, differential operator of zero row sums. Direct application of the, practically convenient, readily applied, Gershgorin’s eigenvalue bounding theorem to this matrix inherently fails to foresee its positive definiteness, pre-dictably, and routinely failing to produce a nontrivial lower bound on the least eigenvalue of this, theoretically assured to be positive definite, matrix. Considered here are practical methods for producing an optimal similarity transformation for the finite-elements global stiffness matrix, following which non trivial, realistic, lower bounds on the least eigenvalue can be located, then further improved. The technique is restricted here to the common case of a global stiffness matrix having only non-positive off-diagonal entries. For such a matrix application of the Gershgorin bounding method may be carried out by a mere matrix vector multiplication.


Introduction
Knowledge, even approximate, of the extremal eigenvalues of the, large, positive definite, global stiffness matrix generated by the finite-element method (see [1]), is essential for assessing the correctness of the numerical solution of the global algebraic system of equations set up by this variational method. Other numerical

Gershgorin's Eigenvalue Bounds
The utterly simple Gershgorin and Rayleigh eigenvalue bound theorems (see [7] [8]) are of such fundamental importance in computational linear algebra that we find it good to fully repeat them here.
Throughout this paper we denote a number by a lower case Greek letter, a vector by a lower case Roman letter, and a matrix by an upper case Roman letter.
The eigenvalue spectrum ( ) A λ of real symmetric matrix A is real. For such a matrix Gershgorin's theorem assumes the simpler form.

( ) T
A A n n A = × = be symmetric, and so of a real eigensystem. Then every eigenvalue of A lies in, or on the end points of, at least one of the intervals

Positive-Definite Matrices with Non-Positive Off-Diagonal Entries
Positive definite and symmetric global stiffness matrices of a positive diagonal and non-negative off-diagonal entries are common in the finite elements, or finite differences, method. For such matrices the application of the Gershgorin theorem simplifies into a mere matrix-vector multiplication operation, efficiently carried out as a summation of such multiplications on the element level of the global mesh (see [6]).
Say matrix A is symmetric, Gershgorin's theorem correctly predicts here that matrix A is, at least, positive semi-definite, but it fails to ascertain that it is actually positive definite, with a positive lowest eigenvalue. To obtain a more realistic, nontrivial, lower bound on the lowest eigenvalue of matrix A we resort to similarity transformations designed to rescue the lower bound from triviality.

Rayleigh Quotient
Let matrix T A A = be real and symmetric. If x is an eigenvector of A for corresponding eigenvalue λ , Ax is the Rayleigh quotient of matrix A. It has some interesting properties of great practical and theoretical reach.
First property: The quotient [ ] R x produces very accurate eigenvalue approximations from even not so good eigenvector approximations. Indeed, let v be an eigenvector corresponding to some λ , and let where w  is the error vector.
The Rayleigh quotient provides inner bounds on all eigenvalues of symmetric matrix A. Hence the Rayleigh bounds and the Gershgorin bounds complement each other.
Theorem (Rayleigh.) Let the eigenvalues of Then with the lower equality holding if and only if Also

Perron's Theorem on Positive Matrices
Positive matrices are common in the finite-element method. The following is a symmetric version of Perron's theorem on positive matrices (see [9]).
Theorem (Perron). If A is a symmetric positive matrix, There can be no other positive vector orthogonal to n x , and so the eigenvector, and also the largest eigenvalue n λ , are unique.

□
The following is another important and useful statement on positive matrices.
Theorem. Suppose that A has a positive inverse and 1 . 1 To prove the other bound write , and have that This completes the proof.

Similarity Transformation, Similar Matrices
We will make also great use of the following fact.
Theorem. If λ is an eigenvalue of matrix A, then λ is also an eigenvalue of similar matrix In this note we are greatly interested in accurate eigenvectors.
To fix ideas consider the typical finite-elements, or finite-differences, matrix The matrix A is symmetric and positive definite, with a corresponding eigenvector that is positive. The fact that 1 A − is completely positive is a manifestation of the physical fact that a point force applied to the system causes all free points of the system to move, and all in the same direction.
To save Gershgorin's theorem from the trivial, and useless prediction ( ) 1 0 A λ ≥ , we propose to similarly transform matrix A with a positive diagonal matrix D, The fact that diagonal matrix D is positive, Namely, the optimal D is such that De is the eigenvector corresponding to the lowest eigenvalue of matrix A. With such a better D, obtained here by inverse iterations on, the here positive, Taking the diagonal entries of matrix D as vector x we obtain from Rayleigh's which are good enclosing bounds.

Taut String (Cable)
The basic issues of this paper are best illustrated on some concrete cases. First we consider the one-dimensional problem of the taut string discretized by n linear finite elements of which the element stiffness matrix is We assemble the element k matrices into the global matrix A, here over a mesh of five elements, impose the essential boundary condition of a fixed end point by deleting the first row and first column of the assembled global matrix, and are left with We know (see [10] [11]) that as the size of matrix ( ) A n n × increases, its lowest eigenvalue decreases, actually

Taut String, Similarity Transformation Followed by a Gershgorin Bound
Being keenly interested in an actual, numerical lower bound on ( )

Taut String, Linearization of the Characteristic Polynomial
We know that as the size of matrix ( ) A n n × increases, its lowest eigenvalue decreases, tending to zero, actually

Taut String, Quadratization of the Characteristic Polynomial
Keeping quadratic terms of λ in the characteristic equation of matrix ( ) we have for vector x of Equation (46) the five quadratically curtailed equations of Ax x λ = : The success of the linearization, or quadratization, of the characteristic equation hinges on the theoretically assured fact (see [10] [11]) that ( )

Taut String, Linearization Following a Shift
To work with a matrix of a lesser minimal eigenvalue we turn to the shifted matrix ( ) ( )

Taut String, Shifted Power Method
We continue with our quest for good approximations to the fundamental eigenvector of the finite elements global stiffness matrix A. We propose to first iterate for the highest eigenvalue and the corresponding eigenvector. We choose to use the power method since it requires but one vector-matrix multiplication per step. We start with

Hanging String
The freely down hanging string (see [12]) is of zero tension at the free lower end, and with the tension growing linearly towards the fixed upper hanging point, that carries the entire weight of the string.
The element stiffness matrix of the hanging string of size h is 1 1 including the essential boundary condition of zero movement of the hanging point. Global stiffness matrix A factors as showing matrix A to be symmetric and positive definite. In fact,

Hanging String, Linearization of the Characteristic Polynomial
Here we have { }

Hanging String, Power Method
The global stiffness matrix we take is still the modest 5 5 ×

Hanging String, Shifted Power Method
Next we turn our attention to computing, an ever better, approximation to ( )

The Four-Nodes Rectangular Membrane Element
Next we move on to two dimensional finite-element membrane problems, or the boundary value problem ( ) with, as expected,

Triangular Membrane, Triangular Finite Elements
The linear, first order, membrane finite element stiffness matrix k for a triangle of sides 1 2 3 , , L L L and area T is ( ) where 1 2 3 , , C C C are the constant matrices Otherwise, the element stiffness matrix is written, by the aid of the law of the cosines, as 2  1  1 2  3  1 3  2  2  1 2  3  2  2 3  1  2  1 3  2  2 3  1  3   cos  cos  1  cos  cos  4 cos cos where i θ is the angle opposite i L . If 2 i θ < π , then 0 ij k < , and so is any global stiffness matrix assembled from it. We consider now an equilateral triangular membrane discretized by four equilateral triangular finite elements per side. The membrane is held fixed on 0 0 0 0 0 2 1 2 6 1 0 0 0 0 0 0 0 1 1 2

Triangular Membrane, Power Method
We start our iterative quest for the highest eigenvalue of matrix A with 0 x , and then continue with the power method to have  entries. The finite element global stiffness matrix readily becomes very large with an ever smaller least eigenvalue.
We have shown here how optimal similarity transformations of this matrix, requiring only matrix-vector multiplication, lead to eminently practical and reliable numerical iterative algorithms for tight realistic Rayleigh and Gershgorin bounds on the least eigenvalue of this large finite-elements global stiffness matrix.