pcg - Solve system of linear equations — preconditioned conjugate gradients
method - MATLAB ([original](https://in.mathworks.com/help/matlab/ref/pcg.html)) ([raw](?raw))Solve system of linear equations — preconditioned conjugate gradients method
Syntax
Description
[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7) = pcg([A](#mw%5Fa05021ac-9e1d-4dd3-84d8-b7d351874e8e),[b](#f93-998351%5Fsep%5Fmw%5F3800dea9-6306-44ef-9490-f16e20ceb7ab)) attempts to solve the system of linear equations A*x = b forx using the Preconditioned Conjugate Gradients Method. When the attempt is successful, pcg displays a message to confirm convergence. Ifpcg fails to converge after the maximum number of iterations or halts for any reason, it displays a diagnostic message that includes the relative residualnorm(b-A*x)/norm(b) and the iteration number at which the method stopped.
[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7) = pcg([A](#mw%5Fa05021ac-9e1d-4dd3-84d8-b7d351874e8e),[b](#f93-998351%5Fsep%5Fmw%5F3800dea9-6306-44ef-9490-f16e20ceb7ab),[tol](#f93-998351%5Fsep%5Fmw%5Fb797b123-ccd8-4b99-a496-07d543d1a21b)) specifies a tolerance for the method. The default tolerance is1e-6.
[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7) = pcg([A](#mw%5Fa05021ac-9e1d-4dd3-84d8-b7d351874e8e),[b](#f93-998351%5Fsep%5Fmw%5F3800dea9-6306-44ef-9490-f16e20ceb7ab),[tol](#f93-998351%5Fsep%5Fmw%5Fb797b123-ccd8-4b99-a496-07d543d1a21b),[maxit](#f93-998351%5Fsep%5Fmw%5F843917a7-f520-49f0-8815-7ffb3f307a16)) specifies the maximum number of iterations to use. pcg displays a diagnostic message if it fails to converge within maxit iterations.
[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7) = pcg([A](#mw%5Fa05021ac-9e1d-4dd3-84d8-b7d351874e8e),[b](#f93-998351%5Fsep%5Fmw%5F3800dea9-6306-44ef-9490-f16e20ceb7ab),[tol](#f93-998351%5Fsep%5Fmw%5Fb797b123-ccd8-4b99-a496-07d543d1a21b),[maxit](#f93-998351%5Fsep%5Fmw%5F843917a7-f520-49f0-8815-7ffb3f307a16),[M](#f93-998351%5Fsep%5Fmw%5Fa88cd7b1-4dcd-46dd-8611-81acb80fe667)) specifies a preconditioner matrix M and computes x by effectively solving the system H−1A H−Ty=H−1b for y, where y=HTx and H=M1/2=(M1M2)1/2. The algorithm does not form H explicitly. Using a preconditioner matrix can improve the numerical properties of the problem and the efficiency of the calculation.
[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7) = pcg([A](#mw%5Fa05021ac-9e1d-4dd3-84d8-b7d351874e8e),[b](#f93-998351%5Fsep%5Fmw%5F3800dea9-6306-44ef-9490-f16e20ceb7ab),[tol](#f93-998351%5Fsep%5Fmw%5Fb797b123-ccd8-4b99-a496-07d543d1a21b),[maxit](#f93-998351%5Fsep%5Fmw%5F843917a7-f520-49f0-8815-7ffb3f307a16),[M1](#f93-998351%5Fsep%5Fmw%5Fa88cd7b1-4dcd-46dd-8611-81acb80fe667),[M2](#f93-998351%5Fsep%5Fmw%5Fa88cd7b1-4dcd-46dd-8611-81acb80fe667)) specifies factors of the preconditioner matrix M such that M = M1*M2.
[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7) = pcg([A](#mw%5Fa05021ac-9e1d-4dd3-84d8-b7d351874e8e),[b](#f93-998351%5Fsep%5Fmw%5F3800dea9-6306-44ef-9490-f16e20ceb7ab),[tol](#f93-998351%5Fsep%5Fmw%5Fb797b123-ccd8-4b99-a496-07d543d1a21b),[maxit](#f93-998351%5Fsep%5Fmw%5F843917a7-f520-49f0-8815-7ffb3f307a16),[M1](#f93-998351%5Fsep%5Fmw%5Fa88cd7b1-4dcd-46dd-8611-81acb80fe667),[M2](#f93-998351%5Fsep%5Fmw%5Fa88cd7b1-4dcd-46dd-8611-81acb80fe667),[x0](#f93-998351%5Fsep%5Fmw%5F40e27791-2264-4d42-8cb6-b8bf869783bc)) specifies an initial guess for the solution vector x. The default is a vector of zeros.
[[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7),[flag](#f93-998351%5Fsep%5Fmw%5F1f278699-812f-4628-8d99-e8f6ae362e34)] = pcg(___) returns a flag that specifies whether the algorithm successfully converged. Whenflag = 0, convergence was successful. You can use this output syntax with any of the previous input argument combinations. When you specify theflag output, pcg does not display any diagnostic messages.
[[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7),[flag](#f93-998351%5Fsep%5Fmw%5F1f278699-812f-4628-8d99-e8f6ae362e34),[relres](#f93-998351%5Fsep%5Fmw%5F6de229ba-091b-4137-acf4-6c7691a0398c)] = pcg(___) also returns the relative residual norm(b-A*x)/norm(b). Ifflag is 0, then relres <= tol.
[[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7),[flag](#f93-998351%5Fsep%5Fmw%5F1f278699-812f-4628-8d99-e8f6ae362e34),[relres](#f93-998351%5Fsep%5Fmw%5F6de229ba-091b-4137-acf4-6c7691a0398c),[iter](#f93-998351%5Fsep%5Fmw%5F98f428f1-5bc3-491c-bdce-7d3acf5e0ed4)] = pcg(___) also returns the iteration number iter at which x was computed.
[[x](#f93-998351%5Fsep%5Fmw%5Fb2afce71-ce1a-494a-85e3-71dd5f36e7d7),[flag](#f93-998351%5Fsep%5Fmw%5F1f278699-812f-4628-8d99-e8f6ae362e34),[relres](#f93-998351%5Fsep%5Fmw%5F6de229ba-091b-4137-acf4-6c7691a0398c),[iter](#f93-998351%5Fsep%5Fmw%5F98f428f1-5bc3-491c-bdce-7d3acf5e0ed4),[resvec](#f93-998351%5Fsep%5Fmw%5Fe2a3240f-74d0-4b34-943e-502f3dd9d174)] = pcg(___) also returns a vector of the residual norm at each iteration, including the first residualnorm(b-A*x0).
Examples
Solve a square linear system using pcg with default settings, and then adjust the tolerance and number of iterations used in the solution process.
Create a random symmetric sparse matrix A. Also create a vector b of the row sums of A for the right-hand side of Ax=b so that the true solution x is a vector of ones.
rng default A = sprand(400,400,.5); A = A'*A; b = sum(A,2);
Solve Ax=b using pcg. The output display includes the value of the relative residual error ‖b-Ax‖‖b‖.
pcg stopped at iteration 20 without converging to the desired tolerance 1e-06 because the maximum number of iterations was reached. The iterate returned (number 20) has relative residual 3.6e-06.
By default pcg uses 20 iterations and a tolerance of 1e-6, and the algorithm is unable to converge in those 20 iterations for this matrix. However, the residual is close to the tolerance, so the algorithm likely just needs more iterations to converge.
Solve the system again using a tolerance of 1e-7 and 150 iterations.
pcg converged at iteration 129 to a solution with relative residual 1e-07.
Examine the effect of using a preconditioner matrix with pcg to solve a linear system.
Create a symmetric positive definite, banded coefficient matrix.
A = delsq(numgrid('S',102));
Define b for the right-hand side of the linear equation Ax=b.
Set the tolerance and maximum number of iterations.
Use pcg to find a solution at the requested tolerance and number of iterations. Specify five outputs to return information about the solution process:
xis the computed solution toA*x = b.fl0is a flag indicating whether the algorithm converged.rr0is the relative residual of the computed answerx.it0is the iteration number whenxwas computed.rv0is a vector of the residual history for ‖b-Ax‖.
[x,fl0,rr0,it0,rv0] = pcg(A,b,tol,maxit); fl0
fl0 is 1 because pcg does not converge to the requested tolerance of 1e-8 within the requested 100 iterations.
To aid with the slow convergence, you can specify a preconditioner matrix. Since A is symmetric, use ichol to generate the preconditioner M=L LT. Solve the preconditioned system by specifying L and L' as inputs to pcg.
L = ichol(A); [x1,fl1,rr1,it1,rv1] = pcg(A,b,tol,maxit,L,L'); fl1
The use of an ichol preconditioner produces a relative residual less than the prescribed tolerance of 1e-8 at the 79th iteration. The output rv1(1) is norm(b) and rv1(end) is norm(b-A*x1).
Now, use the michol option to create a modified incomplete Cholesky preconditioner.
L = ichol(A,struct('michol','on')); [x2,fl2,rr2,it2,rv2] = pcg(A,b,tol,maxit,L,L'); fl2
This preconditioner is better than the one produced by the incomplete Cholesky factorization with zero fill for the coefficient matrix in this example, so pcg is able to converge even quicker.
You can see how the preconditioners affect the rate of convergence of pcg by plotting each of the residual histories starting from the initial estimate (iterate number 0). Add a line for the specified tolerance.
semilogy(0:length(rv0)-1,rv0/norm(b),'-o') hold on semilogy(0:length(rv1)-1,rv1/norm(b),'-o') semilogy(0:length(rv2)-1,rv2/norm(b),'-o') yline(tol,'r--'); legend('No Preconditioner','Default ICHOL','Modified ICHOL','Tolerance','Location','East') xlabel('Iteration number') ylabel('Relative residual')
Examine the effect of supplying pcg with an initial guess of the solution.
Create a tridiagonal sparse matrix. Use the sum of each row as the vector for the right-hand side of Ax=b so that the expected solution for x is a vector of ones.
n = 900; e = ones(n,1); A = spdiags([e 2*e e],-1:1,n,n); b = sum(A,2);
Use pcg to solve Ax=b twice: one time with the default initial guess, and one time with a good initial guess of the solution. Use 200 iterations and the default tolerance for both solutions. Specify the initial guess in the second solution as a vector with all elements equal to 0.99.
maxit = 200; x1 = pcg(A,b,[],maxit);
pcg converged at iteration 35 to a solution with relative residual 9.5e-07.
x0 = 0.99*e; x2 = pcg(A,b,[],maxit,[],[],x0);
pcg converged at iteration 7 to a solution with relative residual 8.7e-07.
In this case supplying an initial guess enables pcg to converge more quickly.
Returning Intermediate Results
You also can use the initial guess to get intermediate results by calling pcg in a for-loop. Each call to the solver performs a few iterations and stores the calculated solution. Then you use that solution as the initial vector for the next batch of iterations.
For example, this code performs 100 iterations four times and stores the solution vector after each pass in the for-loop:
x0 = zeros(size(A,2),1); tol = 1e-8; maxit = 100; for k = 1:4 [x,flag,relres] = pcg(A,b,tol,maxit,[],[],x0); X(:,k) = x; R(k) = relres; x0 = x; end
X(:,k) is the solution vector computed at iteration k of the for-loop, and R(k) is the relative residual of that solution.
Solve a linear system by providing pcg with a function handle that computes A*x in place of the coefficient matrix A.
Use gallery to generate a 20-by-20 positive definite tridiagonal matrix. The super- and subdiagonals have ones, while the main diagonal elements count down from 20 to 1. Preview the matrix.
n = 20; A = gallery('tridiag',ones(n-1,1),n:-1:1,ones(n-1,1)); full(A)
ans = 20×20
20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 19 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 18 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 17 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 16 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 1 15 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 14 1 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 13 1 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 12 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 11 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 1 10 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 1 9 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 1 8 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 1 7 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 6 1 0 0 0 0
⋮Since this tridiagonal matrix has a special structure, you can represent the operation A*x with a function handle. When A multiplies a vector, most of the elements in the resulting vector are zeros. The nonzero elements in the result correspond with the nonzero tridiagonal elements of A. Moreover, only the main diagonal has nonzeros that are not equal to 1.
The expression Ax becomes:
[2010⋯⋯⋯00119100011810⋮⋮011710011610⋮⋮011510011410⋮⋮0113⋱000⋱⋱100⋯⋯⋯011][x1x2x3x4x5⋮⋮x20]=[20x1+x2x1+19x2+x3x2+18x3+x4⋮x18+2x19+x20x19+x20 ].
The resulting vector can be written as the sum of three vectors:
[20x1+x2x1+19x2+x3x2+18x3+x4⋮x18+2x19+x20x19+x20 ]=[0x1⋮x19]+[20x119x2⋮x20]+[x2⋮x200].
In MATLAB®, write a function that creates these vectors and adds them together, thus giving the value of A*x:
function y = afun(x) y = [0; x(1:19)] + ... [(20:-1:1)'].*x + ... [x(2:20); 0]; end
(This function is saved as a local function at the end of the example.)
Now, solve the linear system Ax=b by providing pcg with the function handle that calculates A*x. Use a tolerance of 1e-12 and 50 iterations.
b = ones(20,1);
tol = 1e-12;
maxit = 50;
x1 = pcg(@afun,b,tol,maxit)
pcg converged at iteration 20 to a solution with relative residual 4.4e-16.
x1 = 20×1
0.0476
0.0475
0.0500
0.0526
0.0555
0.0588
0.0625
0.0666
0.0714
0.0769
0.0832
0.0908
0.0998
0.1112
0.1222
⋮Check that afun(x1) produces a vector of ones.
ans = 20×1
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
⋮Local Functions
function y = afun(x) y = [0; x(1:19)] + ... [(20:-1:1)'].*x + ... [x(2:20); 0]; end
Input Arguments
Coefficient matrix, specified as a symmetric positive definite matrix or function handle. This matrix is the coefficient matrix in the linear system A*x = b. Generally, A is a large sparse matrix or a function handle that returns the product of a large sparse matrix and column vector. See Determine Whether Matrix Is Symmetric Positive Definite for info on how to confirm that A is symmetric positive definite.
Specifying A as a Function Handle
You can optionally specify the coefficient matrix as a function handle instead of a matrix. The function handle returns matrix-vector products instead of forming the entire coefficient matrix, making the calculation more efficient.
To use a function handle, use the function signature function y = afun(x). Parameterizing Functions explains how to provide additional parameters to the function afun, if necessary. The function call afun(x) must return the value of A*x.
Data Types: single | double | function_handle
Complex Number Support: Yes
Right side of linear equation, specified as a column vector. The length of b must be equal tosize(A,1).
Data Types: single | double
Complex Number Support: Yes
Method tolerance, specified as a positive scalar. Use this input to trade off accuracy and runtime in the calculation. pcg must meet the tolerance within the number of allowed iterations to be successful. A smaller value of tol means the answer must be more precise for the calculation to be successful.
Data Types: single | double
Maximum number of iterations, specified as a positive scalar integer. Increase the value ofmaxit to allow more iterations forpcg to meet the tolerance tol. Generally, a smaller value of tol means more iterations are required to successfully complete the calculation.
Data Types: single | double
Preconditioner matrices, specified as separate arguments of matrices or function handles. You can specify a preconditioner matrix M or its matrix factors M = M1*M2 to improve the numerical aspects of the linear system and make it easier for pcg to converge quickly. You can use the incomplete matrix factorization functions ilu and ichol to generate preconditioner matrices. You also can use equilibrate prior to factorization to improve the condition number of the coefficient matrix. For more information on preconditioners, see Iterative Methods for Linear Systems.
pcg treats unspecified preconditioners as identity matrices.
Specifying M as a Function Handle
You can optionally specify any of M, M1, orM2 as function handles instead of matrices. The function handle performs matrix-vector operations instead of forming the entire preconditioner matrix, making the calculation more efficient.
To use a function handle, use the function signature function y = mfun(x). Parameterizing Functions explains how to provide additional parameters to the function mfun, if necessary. The function call mfun(x) must return the value ofM\x or M2\(M1\x).
Data Types: single | double | function_handle
Complex Number Support: Yes
Initial guess, specified as a column vector with length equal to size(A,2). If you can provide pcg with a more reasonable initial guessx0 than the default vector of zeros, then it can save computation time and help the algorithm converge faster.
Data Types: single | double
Complex Number Support: Yes
Output Arguments
Linear system solution, returned as a column vector. This output gives the approximate solution to the linear system A*x = b. If the calculation is successful (flag = 0), then relres is less than or equal to tol.
Whenever the calculation is not successful (flag ~= 0), the solutionx returned by pcg is the one with minimal residual norm computed over all the iterations.
Data Types: single | double
Convergence flag, returned as one of the scalar values in this table. The convergence flag indicates whether the calculation was successful and differentiates between several different forms of failure.
| Flag Value | Convergence |
|---|---|
| 0 | Success — pcg converged to the desired tolerance tol withinmaxit iterations. |
| 1 | Failure — pcg iteratedmaxit iterations but did not converge. |
| 2 | Failure — The preconditioner matrix M orM = M1*M2 is ill conditioned. |
| 3 | Failure — pcg stagnated after two consecutive iterations were the same. |
| 4 | Failure — One of the scalar quantities calculated by thepcg algorithm became too small or too large to continue computing. |
Relative residual error, returned as a scalar. The relative residual error relres = norm(b-A*x)/norm(b) is an indication of how accurate the answer is. If the calculation converges to the tolerance tol within maxit iterations, then relres <= tol.
Data Types: single | double
Iteration number, returned as a scalar. This output indicates the iteration number at which the computed answer for x was calculated.
Data Types: double
Residual error, returned as a vector. The residual error norm(b-A*x) reveals how close the algorithm is to converging for a given value ofx. The number of elements in resvec is equal to the number of iterations. You can examine the contents of resvec to help decide whether to change the values of tol ormaxit.
Data Types: single | double
More About
The preconditioned conjugate gradients method (PCG) was developed to exploit the structure of symmetric positive definite matrices. Several other algorithms can operate on symmetric positive definite matrices, but PCG is the quickest and most reliable at solving those types of systems [1].
Tips
- Convergence of most iterative methods depends on the condition number of the coefficient matrix,
cond(A). WhenAis square, you can use equilibrate to improve its condition number, and on its own this makes it easier for most iterative solvers to converge. However, usingequilibratealso leads to better quality preconditioner matrices when you subsequently factor the equilibrated matrixB = R*P*A*C. - You can use matrix reordering functions such as
dissectandsymrcmto permute the rows and columns of the coefficient matrix and minimize the number of nonzeros when the coefficient matrix is factored to generate a preconditioner. This can reduce the memory and time required to subsequently solve the preconditioned linear system.
References
[1] Barrett, R., M. Berry, T. F. Chan, et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
Extended Capabilities
Usage notes and limitations:
- After execution, code generation for
pcgdoes not display a diagnostic message. - You can specify only input
Aas sparse. Other inputs (vector/matrix) must be nonsparse. - You must enable support for variable-size arrays.
- The number of iterations
iterrequired for the algorithm to converge in the generated code might be different from the iteration values in MATLAB®.
The pcg function supports GPU array input with these usage notes and limitations:
- When input
Ais a sparse matrix:- If you use two preconditioners,
M1andM2, then they must be lower triangular and upper triangular matrices, or both of them must be function handles. Using lower triangular and upper triangular preconditioner matrices instead of function handles can significantly improve computation speed. - For GPU arrays,
pcgdoes not detect stagnation (Flag 3). Instead, it reports failure to converge (Flag 1).
- If you use two preconditioners,
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced before R2006a
You can specify these arguments as single precision:
A— Coefficient matrixb— Right side of linear equationM,M1,M2— Preconditioner matricesx0— Initial guess
If you specify any of these arguments as single precision, the function computes in single precision and returns the linear system solution, relative residual error, and residual error outputs as type single. For faster computation, specify all arguments, including function handle outputs, as the same precision.