conjgrad

Arguments

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.

• matrix.If A is a matrix, it can be dense or sparse

• function.If A is a function, it must have the following header :

  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)

method

scalar string, "pcg", "cgs", "bicg" or "bicgstab" (default "bicgstab")

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 M\x (default: none)

M2

preconditioner: full or sparse matrix or function returning M2\x for each x (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 conjgrad converged to the desired tolerance within maxi iterations, 1 else

err

final relative norm of the residual (the 2-norm of the right-hand side b is used)

iter

number of iterations performed

res

vector of the residual relative norms

Description

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 syntax y=Milx(x) which computes the corresponding left division y=Mi\x.

The method input argument selects the solver to be used:

• "pcg" Preconditioned Conjugate Gradient
• "cgs" preconditioned Conjugate Gradient Squared
• "bicg" preconditioned BiConjugate Gradient
• "bicgstab" preconditioned BiConjugate Gradient Stabilized (default)

If method="pcg", then the A matrix must be a symmetric positive definite matrix (full or sparse) or a function with syntax y=Ax(x) which computes y=A*x. Otherwise (method="cgs", "bicg" or "bicgstab"), A just needs to be square.

Example with well-conditioned and ill-conditioned problems

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-conditioned 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] = conjgrad(A, b, "bicg", 1d-12, 15);
mprintf("      fail=%d, iter=%d, errrel=%e\n",fail,iter,err)

// Ill-conditioned 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] = conjgrad(A, b, method="pcg", maxIter=30, tol=1d-12);
mprintf("      fail=%d, iter=%d, errrel=%e\n",fail,iter,err)

Examples with A given as a sparse matrix, or function, or list

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 "conjgrad" primitive. The final case shown by this example, is when a list is passed to the primitive.

// Well-conditioned 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] = conjgrad(Asparse, b, "cgs", 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] = conjgrad(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] = conjgrad(Alist, b, maxIter=30, tol=1d-12);
mprintf("      fail=%d, iter=%d, errrel=%e\n",fail,iter,err)

Examples with key=value syntax

The following example shows how to pass arguments with the "key=value" syntax. This allows to set non-positional 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 positional 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] = conjgrad(A, b, verbose=1);

// With key=value syntax, the order does not matter
[xcomputed, flag, err, iter, res] = conjgrad(A, b, verbose=1, maxIter=0);

See also

• backslash — (\) divisão matricial direita-esquerda: exact or least square solution
• qmr — quasi minimal residual method with preconditioning
• gmres — Generalized Minimum RESidual method

References

PCG

"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

CGS

"CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems" by Peter Sonneveld.

Original article

Article on ACM

Some theory around CGS

BICG

"Numerical Recipes: The Art of Scientific Computing." (third ed.) by William Press, Saul Teukolsky, William Vetterling, Brian Flannery.

http://apps.nrbook.com/empanel/index.html?pg=87

Article on ACM

Some theory around BICG

BICGSTAB

"Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems" by Henk van der Vorst. 339

Original article

Article on ACM

Some theory around BICG

History