pcg: Preconditioned Conjugate Gradient algorithm for solving Ax=b

In pcg: Preconditioned Conjugate Gradient Algorithm for solving Ax=b

Description

The function solves linear system of equations Ax=b by Preconditioned Conjugate Gradient algorithm. Here matrix A must be real symmetric and positive definite. This can also be used to minimize the quadractic function (x'Ax)/2-bx.

Usage

pcg(A, b, M, maxiter = 1e+05, tol = 1e-06)

Arguments

A	A is real symmetric positive definite matrix of order n x n.
b	b is a vector of order n x 1.
M	Optionally a suitable preconditioner matrix specified by user
maxiter	Maximum number of iterations
tol	Tolerance for convergence of the solution

Value

A vector of order n x 1

Note

The algorithm does not check for symmetricity and positive definiteness of matrix A. Please ensure these conditions yourself.

Author(s)

B N Mandal and Jun Ma

References

Barrett, R., M. Berry, T. F. Chan, et al., (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia.

Examples

A=matrix(rnorm(100*100,mean=10,sd=2),100,100)
A=t(A)%*%A
b=rnorm(100)
pcg(A,b)

Example output

               [,1]
  [1,]  -4.96560956
  [2,]   3.05419176
  [3,]   1.63063568
  [4,]  -2.18459432
  [5,]   1.21032034
  [6,]   3.26355283
  [7,]   3.93668599
  [8,]  -9.13796637
  [9,]  11.65146587
 [10,]  -5.23377138
 [11,]  13.93174518
 [12,]   0.16054274
 [13,]   4.73294896
 [14,]   6.50665858
 [15,]   7.10447518
 [16,]   1.79857837
 [17,]  -6.69538837
 [18,]  -1.21559214
 [19,]  -5.92581032
 [20,]  -6.81361844
 [21,]  -2.79639313
 [22,]   1.62021616
 [23,]   3.70009622
 [24,]  -3.19388640
 [25,] -14.05084729
 [26,]  -1.85522249
 [27,]  11.17982821
 [28,]  -1.19679409
 [29,] -11.38436192
 [30,] -15.14559363
 [31,]  17.02627447
 [32,] -13.25885876
 [33,]   0.94089797
 [34,]  -4.09981102
 [35,]   4.32297474
 [36,]   4.46882818
 [37,]  -4.94451999
 [38,] -17.54230745
 [39,]   0.61427055
 [40,]   0.66549476
 [41,]   4.11747453
 [42,]  -2.79753298
 [43,]   6.49912372
 [44,]  -6.76038612
 [45,]  -2.77430641
 [46,]   7.54972018
 [47,]  11.00250595
 [48,]  -3.28564713
 [49,] -11.91920424
 [50,]   0.10118698
 [51,]  -4.22532396
 [52,]   1.97021932
 [53,]   8.96777196
 [54,]  -9.24191298
 [55,]  -6.51947003
 [56,]   4.07824534
 [57,]   1.10269998
 [58,]  12.74205604
 [59,]   7.75945224
 [60,]   1.83902669
 [61,]   6.57911499
 [62,]  -1.46020907
 [63,]   9.39404534
 [64,]  -3.77475572
 [65,]   8.02208607
 [66,]  -0.10143988
 [67,]  -0.09957911
 [68,]  -1.86762261
 [69,]   2.69499642
 [70,]  -4.60228116
 [71,]  -4.15979847
 [72,]  -6.48434535
 [73,]   6.30554143
 [74,]   3.22293433
 [75,]   2.48350616
 [76,]   0.83659425
 [77,]  -5.25165746
 [78,]  -3.40364844
 [79,]   1.73630618
 [80,]   2.55646064
 [81,]   1.64049113
 [82,]   0.64933086
 [83,]  10.10440556
 [84,]  -7.51844897
 [85,]   3.74710674
 [86,]  10.13272057
 [87,]   1.70747368
 [88,]   8.48573246
 [89,] -11.24083905
 [90,]  11.17453306
 [91,]  13.28371648
 [92,] -11.97890216
 [93,]   0.29600168
 [94,]   2.66890762
 [95,]  -9.21662910
 [96,] -13.38643529
 [97,]   1.81879652
 [98,]  -1.09208063
 [99,] -17.18064727
[100,]   5.06407462

pcg documentation built on May 2, 2019, 8:19 a.m.