precondioned conjugate gradient
[x, flag, err, iter, res] = pcg(A, b [, tol [, maxIter [, M [, M2 [, x0 [, verbose]]]]]])
[x, flag, err, iter, res] = pcg(A, b [key=value,...])
:A a matrix, or a function, or a list computing A*x for each given x. The following is a description of the computation of A*x depending on the type of A.
function y=A(x)
+ `list.`If A is a list, the first element of the list is expected to
be a function and the other elements in the list are the arguments of
the function, from index 2 to the end. When the function is called,
the current value of x is passed to the function as the first
argument. The other arguments passed are the one given in the list.
: :b right hand side vector (size: nx1) : :tol error relative tolerance (default: 1e-8). The termination
criteria is based on the 2-norm of the residual r=b-Ax, divided by the 2-norm of the right hand side b.
: :maxIter maximum number of iterations (default: n) : :M preconditioner: full or sparse matrix or function returning Mx
(default: none)
: :x0 initial guess vector (default: zeros(n,1)) : :verbose set to 1 to enable verbose logging (default 0) : :x solution vector : :flag 0 if pcg converged to the desired tolerance within maxi
iterations, 1 else
: :iter number of iterations performed : :res vector of the residual relative norms :
Solves the linear system Ax=b using the conjugate gradient method with or without preconditioning. The preconditionning should be defined by a symmetric positive definite matrix M, or two matrices M1 and M2 such that M=M1*M2. in the case the function solves inv(M)*A*x = inv(M)*b for x. M, M1 and M2 can be Scilab functions with calling sequence y=Milx(x) which computes the corresponding left division y=Mix.
The A matrix must be a symmetric positive definite matrix (full or sparse) or a function with calling sequence y=Ax(x) which computes y=A*x
In the following example, two linear systems are solved. The first maxtrix has a condition number equals to ~0.02, which makes the algorithm converge in exactly 10 iterations. Since this is the size of the matrix, it is an expected behaviour for a gradient conjugate method. The second one has a low condition number equals to 1.d-6, which makes the algorithm converge in a larger 22 iterations. This is why the parameter maxIter is set to 30. See below for other examples of the “key=value” syntax.
//Well conditionned problem
A=[ 94 0 0 0 0 28 0 0 32 0
0 59 13 5 0 0 0 10 0 0
0 13 72 34 2 0 0 0 0 65
0 5 34 114 0 0 0 0 0 55
0 0 2 0 70 0 28 32 12 0
28 0 0 0 0 87 20 0 33 0
0 0 0 0 28 20 71 39 0 0
0 10 0 0 32 0 39 46 8 0
32 0 0 0 12 33 0 8 82 11
0 0 65 55 0 0 0 0 11 100];
b=`ones`_(10,1);
[x, fail, err, iter, res]=pcg(A,b,1d-12,15);
`mprintf`_(" fail=%d, iter=%d, errrel=%e\n",fail,iter,err)
//Ill contionned one
A=[ 894 0 0 0 0 28 0 0 1000 70000
0 5 13 5 0 0 0 0 0 0
0 13 72 34 0 0 0 0 0 6500
0 5 34 1 0 0 0 0 0 55
0 0 0 0 70 0 28 32 12 0
28 0 0 0 0 87 20 0 33 0
0 0 0 0 28 20 71 39 0 0
0 0 0 0 32 0 39 46 8 0
1000 0 0 0 12 33 0 8 82 11
70000 0 6500 55 0 0 0 0 11 100];
[x, fail, err, iter, res]=pcg(A,b,maxIter=30,tol=1d-12);
`mprintf`_(" fail=%d, iter=%d, errrel=%e\n",fail,iter,err)
The following example shows that the method can handle sparse matrices as well. It also shows the case where a function, computing the right- hand side, is given to the “pcg” primitive. The final case shown by this example, is when a list is passed to the primitive.
//Well conditionned problem
A=[ 94 0 0 0 0 28 0 0 32 0
0 59 13 5 0 0 0 10 0 0
0 13 72 34 2 0 0 0 0 65
0 5 34 114 0 0 0 0 0 55
0 0 2 0 70 0 28 32 12 0
28 0 0 0 0 87 20 0 33 0
0 0 0 0 28 20 71 39 0 0
0 10 0 0 32 0 39 46 8 0
32 0 0 0 12 33 0 8 82 11
0 0 65 55 0 0 0 0 11 100];
b=`ones`_(10,1);
// Convert A into a sparse matrix
Asparse=`sparse`_(A);
[x, fail, err, iter, res]=pcg(Asparse,b,maxIter=30,tol=1d-12);
`mprintf`_(" fail=%d, iter=%d, errrel=%e\n",fail,iter,err)
// Define a function which computes the right-hand side.
function y=Atimesx(x)
A=[ 94 0 0 0 0 28 0 0 32 0
0 59 13 5 0 0 0 10 0 0
0 13 72 34 2 0 0 0 0 65
0 5 34 114 0 0 0 0 0 55
0 0 2 0 70 0 28 32 12 0
28 0 0 0 0 87 20 0 33 0
0 0 0 0 28 20 71 39 0 0
0 10 0 0 32 0 39 46 8 0
32 0 0 0 12 33 0 8 82 11
0 0 65 55 0 0 0 0 11 100];
y=A*x
endfunction
// Pass the script Atimesx to the primitive
[x, fail, err, iter, res]=pcg(Atimesx,b,maxIter=30,tol=1d-12);
`mprintf`_(" fail=%d, iter=%d, errrel=%e\n",fail,iter,err)
// Define a function which computes the right-hand side.
function y=Atimesxbis(x, A)
y=A*x
endfunction
// Pass a list to the primitive
Alist = `list`_(Atimesxbis,Asparse);
[x, fail, err, iter, res]=pcg(Alist,b,maxIter=30,tol=1d-12);
`mprintf`_(" fail=%d, iter=%d, errrel=%e\n",fail,iter,err)
The following example shows how to pass arguments with the “key=value” syntax. This allows to set non-positionnal arguments, that is, to set arguments which are not depending on their order in the list of arguments. The available keys are the names of the optional arguments, that is : tol, maxIter, %M, %M2, x0, verbose. Notice that, in the following example, the verbose option is given before the maxIter option. Without the “key=value” syntax, the positionnal arguments would require that maxIter come first and verbose after.
// Example of an argument passed with key=value syntax
A=[100,1;1,10];
b=[101;11];
[xcomputed, flag, err, iter, res]=pcg(A,b,verbose=1);
// With key=value syntax, the order does not matter
[xcomputed, flag, err, iter, res]=pcg(A,b,verbose=1,maxIter=0);
“Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods”, Barrett, Berry, Chan, Demmel, Donato, Dongarra, Eijkhout, Pozo, Romine, and Van der Vorst, SIAM Publications, 1993, ftp netlib2.cs.utk.edu/linalg/templates.ps
“Iterative Methods for Sparse Linear Systems, Second Edition”, Saad, SIAM Publications, 2003, ftp ftp.cs.umn.edu/dept/users/saad/PS/all_ps.zip