Conjugate gradient method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems.
The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched.
Description of the problem addressed by conjugate gradients
Suppose we want to solve the system of linear equations
for the vector , where the known matrix is symmetric (i.e., AT = A), positive-definite (i.e. xTAx > 0 for all non-zero vectors in Rn), and real, and is known as well. We denote the unique solution of this system by .
Derivation as a direct method
The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. Despite differences in their approaches, these derivations share a common topic—proving the orthogonality of the residuals and conjugacy of the search directions. These two properties are crucial to developing the well-known succinct formulation of the method.
We say that two non-zero vectors u and v are conjugate (with respect to ) if
Since is symmetric and positive-definite, the left-hand side defines an inner product
Two vectors are conjugate if and only if they are orthogonal with respect to this inner product. Being conjugate is a symmetric relation: if is conjugate to , then is conjugate to . Suppose that
is a set of mutually conjugate vectors with respect to , i.e. for all . Then forms a basis for , and we may express the solution of in this basis:
Left-multiplying by yields
This gives the following method for solving the equation Ax = b: find a sequence of conjugate directions, and then compute the coefficients .
As an iterative method
If we choose the conjugate vectors carefully, then we may not need all of them to obtain a good approximation to the solution . So, we want to regard the conjugate gradient method as an iterative method. This also allows us to approximately solve systems where n is so large that the direct method would take too much time.
We denote the initial guess for x∗ by x0 (we can assume without loss of generality that x0 = 0, otherwise consider the system Az = b − Ax0 instead). Starting with x0 we search for the solution and in each iteration we need a metric to tell us whether we are closer to the solution x∗ (that is unknown to us). This metric comes from the fact that the solution x∗ is also the unique minimizer of the following quadratic function
The existence of a unique minimizer is apparent as its Hessian matrix of second derivatives is symmetric positive-definite
and that the minimizer (use Df(x)=0) solves the initial problem is obvious from its first derivative
This suggests taking the first basis vector p0 to be the negative of the gradient of f at x = x0. The gradient of f equals Ax − b. Starting with an initial guess x0, this means we take p0 = b − Ax0. The other vectors in the basis will be conjugate to the gradient, hence the name conjugate gradient method. Note that p0 is also the residual provided by this initial step of the algorithm.
Let rk be the residual at the kth step:
As observed above, is the negative gradient of at , so the gradient descent method would require to move in the direction rk. Here, however, we insist that the directions be conjugate to each other. A practical way to enforce this is by requiring that the next search direction be built out of the current residual and all previous search directions. The conjugation constraint is an orthonormal-type constraint and hence the algorithm can be viewed as an example of Gram-Schmidt orthonormalization. This gives the following expression:
(see the picture at the top of the article for the effect of the conjugacy constraint on convergence). Following this direction, the next optimal location is given by
where the last equality follows from the definition of . The expression for can be derived if one substitutes the expression for xk+1 into f and minimizing it w.r.t.
The resulting algorithm
The above algorithm gives the most straightforward explanation of the conjugate gradient method. Seemingly, the algorithm as stated requires storage of all previous searching directions and residue vectors, as well as many matrix-vector multiplications, and thus can be computationally expensive. However, a closer analysis of the algorithm shows that is orthogonal to , i.e. , for i ≠ j. And is -orthogonal to , i.e. , for . This can be regarded that as the algorithm progresses, and span the same Krylov subspace. Where form the orthogonal basis with respect to the standard inner product, and form the orthogonal basis with respect to the inner product induced by . Therefore, can be regarded as the projection of on the Krylov subspace.
The algorithm is detailed below for solving Ax = b where is a real, symmetric, positive-definite matrix. The input vector can be an approximate initial solution or 0. It is a different formulation of the exact procedure described above.
This is the most commonly used algorithm. The same formula for βk is also used in the Fletcher–Reeves nonlinear conjugate gradient method.
We note that is computed by the gradient descent method applied to . Setting would similarly make computed by the gradient descent method from , i.e., can be used as a simple implementation of a restart of the conjugate gradient iterations. Restarts could slow down convergence, but may improve stability if the conjugate gradient method misbehaves, e.g., due to round-off error.
Explicit residual calculation
The formulas and , which both hold in exact arithmetic, make the formulas and mathematically equivalent. The former is used in the algorithm to avoid an extra multiplication by since the vector is already computed to evaluate . The latter may be more accurate, substituting the explicit calculation for the implicit one by the recursion subject to round-off error accumulation, and is thus recommended for an occasional evaluation.
A norm of the residual is typically used for stopping criteria. The norm of the explicit residual provides a guaranteed level of accuracy both in exact arithmetic and in the presence of the rounding errors, where convergence naturally stagnates. In contrast, the implicit residual is known to keep getting smaller in amplitude well below the level of rounding errors and thus cannot be used to determine the stagnation of convergence.
Computation of alpha and beta
In the algorithm, αk is chosen such that is orthogonal to . The denominator is simplified from
since . The βk is chosen such that is conjugate to . Initially, βk is
the numerator of βk is rewritten as
because and are orthogonal by design. The denominator is rewritten as
using that the search directions pk are conjugated and again that the residuals are orthogonal. This gives the β in the algorithm after cancelling αk.
Example code in MATLAB / GNU Octave
function x = conjgrad(A, b, x) r = b - A * x; p = r; rsold = r' * r; for i = 1:length(b) Ap = A * p; alpha = rsold / (p' * Ap); x = x + alpha * p; r = r - alpha * Ap; rsnew = r' * r; if sqrt(rsnew) < 1e-10 break end p = r + (rsnew / rsold) * p; rsold = rsnew; end end
Consider the linear system Ax = b given by
we will perform two steps of the conjugate gradient method beginning with the initial guess
in order to find an approximate solution to the system.
For reference, the exact solution is
Our first step is to calculate the residual vector r0 associated with x0. This residual is computed from the formula r0 = b - Ax0, and in our case is equal to
Since this is the first iteration, we will use the residual vector r0 as our initial search direction p0; the method of selecting pk will change in further iterations.
We now compute the scalar α0 using the relationship
We can now compute x1 using the formula
This result completes the first iteration, the result being an "improved" approximate solution to the system, x1. We may now move on and compute the next residual vector r1 using the formula
Our next step in the process is to compute the scalar β0 that will eventually be used to determine the next search direction p1.
Now, using this scalar β0, we can compute the next search direction p1 using the relationship
We now compute the scalar α1 using our newly acquired p1 using the same method as that used for α0.
Finally, we find x2 using the same method as that used to find x1.
The result, x2, is a "better" approximation to the system's solution than x1 and x0. If exact arithmetic were to be used in this example instead of limited-precision, then the exact solution would theoretically have been reached after n = 2 iterations (n being the order of the system).
The conjugate gradient method can theoretically be viewed as a direct method, as in the absence of round-off error it produces the exact solution after a finite number of iterations, which is not larger than the size of the matrix. In practice, the exact solution is never obtained since the conjugate gradient method is unstable with respect to even small perturbations, e.g., most directions are not in practice conjugate, due to a degenerative nature of generating the Krylov subspaces.
As an iterative method, the conjugate gradient method monotonically (in the energy norm) improves approximations to the exact solution and may reach the required tolerance after a relatively small (compared to the problem size) number of iterations. The improvement is typically linear and its speed is determined by the condition number of the system matrix : the larger is, the slower the improvement.
If is large, preconditioning is commonly used to replace the original system with such that is smaller than , see below.
Define a subset of polynomials as
where is the set of polynomials of maximal degree .
Note, the important limit when tends to
If initialized randomly, the first stage of iterations is often the fastest, as the error is eliminated within the Krylov subspace that initially reflects a smaller effective condition number. The second stage of convergence is typically well defined by the theoretical convergence bound with , but may be super-linear, depending on a distribution of the spectrum of the matrix and the spectral distribution of the error. In the last stage, the smallest attainable accuracy is reached and the convergence stalls or the method may even start diverging. In typical scientific computing applications in double-precision floating-point format for matrices of large sizes, the conjugate gradient method uses a stopping criteria with a tolerance that terminates the iterations during the first or second stage.
The preconditioned conjugate gradient method
- if rk+1 is sufficiently small then exit loop end if
- end repeat
- The result is xk+1
The above formulation is equivalent to applying the conjugate gradient method without preconditioning to the system
The preconditioner matrix M has to be symmetric positive-definite and fixed, i.e., cannot change from iteration to iteration. If any of these assumptions on the preconditioner is violated, the behavior of the preconditioned conjugate gradient method may become unpredictable.
The flexible preconditioned conjugate gradient method
In numerically challenging applications, sophisticated preconditioners are used, which may lead to variable preconditioning, changing between iterations. Even if the preconditioner is symmetric positive-definite on every iteration, the fact that it may change makes the arguments above invalid, and in practical tests leads to a significant slow down of the convergence of the algorithm presented above. Using the Polak–Ribière formula
instead of the Fletcher–Reeves formula
may dramatically improve the convergence in this case. This version of the preconditioned conjugate gradient method can be called flexible, as it allows for variable preconditioning. The flexible version is also shown to be robust even if the preconditioner is not symmetric positive definite (SPD).
The implementation of the flexible version requires storing an extra vector. For a fixed SPD preconditioner, so both formulas for βk are equivalent in exact arithmetic, i.e., without the round-off error.
The mathematical explanation of the better convergence behavior of the method with the Polak–Ribière formula is that the method is locally optimal in this case, in particular, it does not converge slower than the locally optimal steepest descent method.
Vs. the locally optimal steepest descent method
In both the original and the preconditioned conjugate gradient methods one only needs to set in order to make them locally optimal, using the line search, steepest descent methods. With this substitution, vectors p are always the same as vectors z, so there is no need to store vectors p. Thus, every iteration of these steepest descent methods is a bit cheaper compared to that for the conjugate gradient methods. However, the latter converge faster, unless a (highly) variable and/or non-SPD preconditioner is used, see above.
Conjugate gradient method as optimal feedback controller for double integrator
Conjugate gradient on the normal equations
The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations ATA and right-hand side vector ATb, since ATA is a symmetric positive-semidefinite matrix for any A. The result is conjugate gradient on the normal equations (CGNR).
- ATAx = ATb
As an iterative method, it is not necessary to form ATA explicitly in memory but only to perform the matrix-vector and transpose matrix-vector multiplications. Therefore, CGNR is particularly useful when A is a sparse matrix since these operations are usually extremely efficient. However the downside of forming the normal equations is that the condition number κ(ATA) is equal to κ2(A) and so the rate of convergence of CGNR may be slow and the quality of the approximate solution may be sensitive to roundoff errors. Finding a good preconditioner is often an important part of using the CGNR method.
Conjugate gradient method for complex Hermitian matrices
The conjugate gradient method with a trivial modification is extendable to solving, given complex-valued matrix A and vector b, the system of linear equations for the complex-valued vector x, where A is Hermitian (i.e., A' = A) and positive-definite matrix, and the symbol ' denotes the conjugate transpose using the MATLAB/GNU Octave style. The trivial modification is simply substituting the conjugate transpose for the real transpose everywhere. This substitution is backward compatible, since conjugate transpose turns into real transpose on real-valued vectors and matrices. The provided above Example code in MATLAB/GNU Octave thus already works for complex Hermitian matrices needed no modification.
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