The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power methods to find the m "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an n \times n Hermitian matrix, where m is often but not necessarily much smaller than n. Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability. In 1970, Ojalvo and Newman showed how to make the method numerically stable and applied it to the solution of very large engineering structures subjected to dynamic loading. This was achieved using a method for purifying the Lanczos vectors (i.e. by repeatedly reorthogonalizing each newly generated vector with all previously generated ones) to any degree of accuracy, which when not performed, produced a series of vectors that were highly contaminated by those associated with the lowest natural frequencies. In their original work, these authors also suggested how to select a starting vector (i.e. use a random-number generator to select each element of the starting vector) and suggested an empirically determined method for determining m, the reduced number of vectors (i.e. it should be selected to be approximately 1.5 times the number of accurate eigenvalues desired). Soon thereafter their work was followed by Paige, who also provided an error analysis. In 1988, Ojalvo produced a more detailed history of this algorithm and an efficient eigenvalue error test.

The algorithm

There are in principle four ways to write the iteration procedure. Paige and other works show that the above order of operations is the most numerically stable. In practice the initial vector v_1 may be taken as another argument of the procedure, with \beta_j=0 and indicators of numerical imprecision being included as additional loop termination conditions. Not counting the matrix–vector multiplication, each iteration does O(n) arithmetical operations. The matrix–vector multiplication can be done in O(dn) arithmetical operations where d is the average number of nonzero elements in a row. The total complexity is thus O(dmn), or O(dn^2) if m=n; the Lanczos algorithm can be very fast for sparse matrices. Schemes for improving numerical stability are typically judged against this high performance. The vectors v_j are called Lanczos vectors. The vector w_j' is not used after w_j is computed, and the vector w_j is not used after v_{j+1} is computed. Hence one may use the same storage for all three. Likewise, if only the tridiagonal matrix T is sought, then the raw iteration does not need v_{j-1} after having computed w_j, although some schemes for improving the numerical stability would need it later on. Sometimes the subsequent Lanczos vectors are recomputed from v_1 when needed.

Application to the eigenproblem

The Lanczos algorithm is most often brought up in the context of finding the eigenvalues and eigenvectors of a matrix, but whereas an ordinary diagonalization of a matrix would make eigenvectors and eigenvalues apparent from inspection, the same is not true for the tridiagonalization performed by the Lanczos algorithm; nontrivial additional steps are needed to compute even a single eigenvalue or eigenvector. Nonetheless, applying the Lanczos algorithm is often a significant step forward in computing the eigendecomposition. If \lambda is an eigenvalue of T, and x its eigenvector, then y = V x is a corresponding eigenvector of A with the same eigenvalue: Thus the Lanczos algorithm transforms the eigendecomposition problem for A into the eigendecomposition problem for T. The combination of good performance for sparse matrices and the ability to compute several (without computing all) eigenvalues are the main reasons for choosing to use the Lanczos algorithm.

Application to tridiagonalization

Though the eigenproblem is often the motivation for applying the Lanczos algorithm, the operation the algorithm primarily performs is tridiagonalization of a matrix, for which numerically stable Householder transformations have been favoured since the 1950s. During the 1960s the Lanczos algorithm was disregarded. Interest in it was rejuvenated by the Kaniel–Paige convergence theory and the development of methods to prevent numerical instability, but the Lanczos algorithm remains the alternative algorithm that one tries only if Householder is not satisfactory. Aspects in which the two algorithms differ include:

Derivation of the algorithm

There are several lines of reasoning which lead to the Lanczos algorithm.

A more provident power method

The power method for finding the eigenvalue of largest magnitude and a corresponding eigenvector of a matrix A is roughly A critique that can be raised against this method is that it is wasteful: it spends a lot of work (the matrix–vector products in step 2.1) extracting information from the matrix A, but pays attention only to the very last result; implementations typically use the same variable for all the vectors u_j, having each new iteration overwrite the results from the previous one. It may be desirable to instead keep all the intermediate results and organise the data. One piece of information that trivially is available from the vectors u_j is a chain of Krylov subspaces. One way of stating that without introducing sets into the algorithm is to claim that it computes this is trivially satisfied by v_j = u_j as long as u_j is linearly independent of (and in the case that there is such a dependence then one may continue the sequence by picking as v_j an arbitrary vector linearly independent of ). A basis containing the u_j vectors is however likely to be numerically ill-conditioned, since this sequence of vectors is by design meant to converge to an eigenvector of A. To avoid that, one can combine the power iteration with a Gram–Schmidt process, to instead produce an orthonormal basis of these Krylov subspaces. The relation between the power iteration vectors u_j and the orthogonal vectors v_j is that Here it may be observed that we do not actually need the u_j vectors to compute these v_j, because and therefore the difference between and is in, which is cancelled out by the orthogonalisation process. Thus the same basis for the chain of Krylov subspaces is computed by A priori the coefficients h_{k,j} satisfy the definition may seem a bit odd, but fits the general pattern since Because the power iteration vectors u_j that were eliminated from this recursion satisfy the vectors and coefficients h_{k,j} contain enough information from A that all of can be computed, so nothing was lost by switching vectors. (Indeed, it turns out that the data collected here give significantly better approximations of the largest eigenvalue than one gets from an equal number of iterations in the power method, although that is not necessarily obvious at this point.) This last procedure is the Arnoldi iteration. The Lanczos algorithm then arises as the simplification one gets from eliminating calculation steps that turn out to be trivial when A is Hermitian—in particular most of the h_{k,j} coefficients turn out to be zero. Elementarily, if A is Hermitian then For k < j-1 we know that, and since v_j by construction is orthogonal to this subspace, this inner product must be zero. (This is essentially also the reason why sequences of orthogonal polynomials can always be given a three-term recurrence relation.) For k = j-1 one gets since the latter is real on account of being the norm of a vector. For k = j one gets meaning this is real too. More abstractly, if V is the matrix with columns then the numbers h_{k,j} can be identified as elements of the matrix H = V^*AV, and h_{k,j} = 0 for k > j+1; the matrix H is upper Hessenberg. Since the matrix H is Hermitian. This implies that H is also lower Hessenberg, so it must in fact be tridiagional. Being Hermitian, its main diagonal is real, and since its first subdiagonal is real by construction, the same is true for its first superdiagonal. Therefore, H is a real, symmetric matrix—the matrix T of the Lanczos algorithm specification.

Simultaneous approximation of extreme eigenvalues

One way of characterising the eigenvectors of a Hermitian matrix A is as stationary points of the Rayleigh quotient In particular, the largest eigenvalue is the global maximum of r and the smallest eigenvalue is the global minimum of r. Within a low-dimensional subspace \mathcal{L} of \Complex^n it can be feasible to locate the maximum x and minimum y of r. Repeating that for an increasing chain produces two sequences of vectors: and such that and The question then arises how to choose the subspaces so that these sequences converge at optimal rate. From x_j, the optimal direction in which to seek larger values of r is that of the gradient, and likewise from y_j the optimal direction in which to seek smaller values of r is that of the negative gradient. In general so the directions of interest are easy enough to compute in matrix arithmetic, but if one wishes to improve on both x_j and y_j then there are two new directions to take into account: Ax_j and Ay_j; since x_j and y_j can be linearly independent vectors (indeed, are close to orthogonal), one cannot in general expect Ax_j and Ay_j to be parallel. It is not necessary to increase the dimension of by 2 on every step if are taken to be Krylov subspaces, because then for all thus in particular for both z = x_j and z = y_j. In other words, we can start with some arbitrary initial vector x_1 = y_1, construct the vector spaces and then seek such that Since the jth power method iterate u_j belongs to it follows that an iteration to produce the x_j and y_j cannot converge slower than that of the power method, and will achieve more by approximating both eigenvalue extremes. For the subproblem of optimising r on some, it is convenient to have an orthonormal basis for this vector space. Thus we are again led to the problem of iteratively computing such a basis for the sequence of Krylov subspaces.

Convergence and other dynamics

When analysing the dynamics of the algorithm, it is convenient to take the eigenvalues and eigenvectors of A as given, even though they are not explicitly known to the user. To fix notation, let be the eigenvalues (these are known to all be real, and thus possible to order) and let be an orthonormal set of eigenvectors such that for all. It is also convenient to fix a notation for the coefficients of the initial Lanczos vector v_1 with respect to this eigenbasis; let for all, so that. A starting vector v_1 depleted of some eigencomponent will delay convergence to the corresponding eigenvalue, and even though this just comes out as a constant factor in the error bounds, depletion remains undesirable. One common technique for avoiding being consistently hit by it is to pick v_1 by first drawing the elements randomly according to the same normal distribution with mean 0 and then rescale the vector to norm 1. Prior to the rescaling, this causes the coefficients d_k to also be independent normally distributed stochastic variables from the same normal distribution (since the change of coordinates is unitary), and after rescaling the vector will have a uniform distribution on the unit sphere in. This makes it possible to bound the probability that for example. The fact that the Lanczos algorithm is coordinate-agnostic – operations only look at inner products of vectors, never at individual elements of vectors – makes it easy to construct examples with known eigenstructure to run the algorithm on: make A a diagonal matrix with the desired eigenvalues on the diagonal; as long as the starting vector v_1 has enough nonzero elements, the algorithm will output a general tridiagonal symmetric matrix as T.

Kaniel–Paige convergence theory

After m iteration steps of the Lanczos algorithm, T is an m \times m real symmetric matrix, that similarly to the above has m eigenvalues By convergence is primarily understood the convergence of \theta_1 to \lambda_1 (and the symmetrical convergence of \theta_m to \lambda_n) as m grows, and secondarily the convergence of some range of eigenvalues of T to their counterparts of A. The convergence for the Lanczos algorithm is often orders of magnitude faster than that for the power iteration algorithm. The bounds for \theta_1 come from the above interpretation of eigenvalues as extreme values of the Rayleigh quotient r(x). Since \lambda_1 is a priori the maximum of r on the whole of \Complex^n, whereas \theta_1 is merely the maximum on an m-dimensional Krylov subspace, we trivially get. Conversely, any point x in that Krylov subspace provides a lower bound r(x) for \theta_1, so if a point can be exhibited for which is small then this provides a tight bound on \theta_1. The dimension m Krylov subspace is so any element of it can be expressed as p(A) v_1 for some polynomial p of degree at most m-1; the coefficients of that polynomial are simply the coefficients in the linear combination of the vectors. The polynomial we want will turn out to have real coefficients, but for the moment we should allow also for complex coefficients, and we will write p^* for the polynomial obtained by complex conjugating all coefficients of p. In this parametrisation of the Krylov subspace, we have Using now the expression for v_1 as a linear combination of eigenvectors, we get and more generally for any polynomial q. Thus A key difference between numerator and denominator here is that the k=1 term vanishes in the numerator, but not in the denominator. Thus if one can pick p to be large at \lambda_1 but small at all other eigenvalues, one will get a tight bound on the error. Since A has many more eigenvalues than p has coefficients, this may seem a tall order, but one way to meet it is to use Chebyshev polynomials. Writing c_k for the degree k Chebyshev polynomial of the first kind (that which satisfies for all x), we have a polynomial which stays in the range [-1,1] on the known interval [-1,1] but grows rapidly outside it. With some scaling of the argument, we can have it map all eigenvalues except \lambda_1 into [-1,1]. Let (in case, use instead the largest eigenvalue strictly less than \lambda_1), then the maximal value of for is 1 and the minimal value is 0, so Furthermore the quantity (i.e., the ratio of the first eigengap to the diameter of the rest of the spectrum) is thus of key importance for the convergence rate here. Also writing we may conclude that The convergence rate is thus controlled chiefly by R, since this bound shrinks by a factor R^{-2} for each extra iteration. For comparison, one may consider how the convergence rate of the power method depends on \rho, but since the power method primarily is sensitive to the quotient between absolute values of the eigenvalues, we need for the eigengap between \lambda_1 and \lambda_2 to be the dominant one. Under that constraint, the case that most favours the power method is that, so consider that. Late in the power method, the iteration vector: where each new iteration effectively multiplies the z_2-amplitude t by The estimate of the largest eigenvalue is then so the above bound for the Lanczos algorithm convergence rate should be compared to which shrinks by a factor of for each iteration. The difference thus boils down to that between 1+2\rho and. In the \rho \gg 1 region, the latter is more like 1+4\rho, and performs like the power method would with an eigengap twice as large; a notable improvement. The more challenging case is however that of \rho \ll 1, in which is an even larger improvement on the eigengap; the \rho \gg 1 region is where the Lanczos algorithm convergence-wise makes the smallest improvement on the power method.

Numerical stability

Stability means how much the algorithm will be affected (i.e. will it produce the approximate result close to the original one) if there are small numerical errors introduced and accumulated. Numerical stability is the central criterion for judging the usefulness of implementing an algorithm on a computer with roundoff. For the Lanczos algorithm, it can be proved that with exact arithmetic, the set of vectors constructs an orthonormal basis, and the eigenvalues/vectors solved are good approximations to those of the original matrix. However, in practice (as the calculations are performed in floating point arithmetic where inaccuracy is inevitable), the orthogonality is quickly lost and in some cases the new vector could even be linearly dependent on the set that is already constructed. As a result, some of the eigenvalues of the resultant tridiagonal matrix may not be approximations to the original matrix. Therefore, the Lanczos algorithm is not very stable. Users of this algorithm must be able to find and remove those "spurious" eigenvalues. Practical implementations of the Lanczos algorithm go in three directions to fight this stability issue:

Variations

Variations on the Lanczos algorithm exist where the vectors involved are tall, narrow matrices instead of vectors and the normalizing constants are small square matrices. These are called "block" Lanczos algorithms and can be much faster on computers with large numbers of registers and long memory-fetch times. Many implementations of the Lanczos algorithm restart after a certain number of iterations. One of the most influential restarted variations is the implicitly restarted Lanczos method, which is implemented in ARPACK. This has led into a number of other restarted variations such as restarted Lanczos bidiagonalization. Another successful restarted variation is the Thick-Restart Lanczos method, which has been implemented in a software package called TRLan.

Nullspace over a finite field

In 1995, Peter Montgomery published an algorithm, based on the Lanczos algorithm, for finding elements of the nullspace of a large sparse matrix over GF(2); since the set of people interested in large sparse matrices over finite fields and the set of people interested in large eigenvalue problems scarcely overlap, this is often also called the block Lanczos algorithm without causing unreasonable confusion.

Applications

Lanczos algorithms are very attractive because the multiplication by A, is the only large-scale linear operation. Since weighted-term text retrieval engines implement just this operation, the Lanczos algorithm can be applied efficiently to text documents (see latent semantic indexing). Eigenvectors are also important for large-scale ranking methods such as the HITS algorithm developed by Jon Kleinberg, or the PageRank algorithm used by Google. Lanczos algorithms are also used in condensed matter physics as a method for solving Hamiltonians of strongly correlated electron systems, as well as in shell model codes in nuclear physics.

Implementations

The NAG Library contains several routines for the solution of large scale linear systems and eigenproblems which use the Lanczos algorithm. MATLAB and GNU Octave come with ARPACK built-in. Both stored and implicit matrices can be analyzed through the eigs function (Matlab/Octave). Similarly, in Python, the SciPy package has scipy.sparse.linalg.eigsh which is also a wrapper for the SSEUPD and DSEUPD functions functions from ARPACK which use the Implicitly Restarted Lanczos Method. A Matlab implementation of the Lanczos algorithm (note precision issues) is available as a part of the Gaussian Belief Propagation Matlab Package. The GraphLab collaborative filtering library incorporates a large scale parallel implementation of the Lanczos algorithm (in C++) for multicore. The PRIMME library also implements a Lanczos-like algorithm.