org.apache.commons.math3.linear

## Class ConjugateGradient

- java.lang.Object
- - org.apache.commons.math3.linear.IterativeLinearSolver
  - - org.apache.commons.math3.linear.PreconditionedIterativeLinearSolver
    - - org.apache.commons.math3.linear.ConjugateGradient

-     public class ConjugateGradient
      extends PreconditionedIterativeLinearSolver

  This is an implementation of the conjugate gradient method for `RealLinearOperator`. It follows closely the template by Barrett et al. (1994) (figure 2.5). The linear system at hand is A · x = b, and the residual is r = b - A · x.

  ### Default stopping criterion

  A default stopping criterion is implemented. The iterations stop when || r || ≤ δ || b ||, where b is the right-hand side vector, r the current estimate of the residual, and δ a user-specified tolerance. It should be noted that r is the so-called updated residual, which might differ from the true residual due to rounding-off errors (see e.g. Strakos and Tichy, 2002).

  ### Iteration count

  In the present context, an iteration should be understood as one evaluation of the matrix-vector product A · x. The initialization phase therefore counts as one iteration.

  ### Exception context

  Besides standard `DimensionMismatchException`, this class might throw `NonPositiveDefiniteOperatorException` if the linear operator or the preconditioner are not positive definite. In this case, the `ExceptionContext` provides some more information

  - key `"operator"` points to the offending linear operator, say L,
  - key `"vector"` points to the offending vector, say x, such that xT · L · x < 0.

  ### References

  Barret et al. (1994)
    R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. M. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine and H. Van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM
  Strakos and Tichy (2002)
    Z. Strakos and P. Tichy, On error estimation in the conjugate gradient method and why it works in finite precision computations, Electronic Transactions on Numerical Analysis 13: 56-80, 2002
  Since:
    3.0

- - ### Field Summary

    Fields
    Modifier and Type | Field and Description
    --- | ---
    `static String` | `OPERATOR` Key for the exception context.
    `static String` | `VECTOR` Key for the exception context.

  - ### Constructor Summary

    Constructors
    Constructor and Description
    `ConjugateGradient(int maxIterations, double delta, boolean check)` Creates a new instance of this class, with default stopping criterion.
    `ConjugateGradient(IterationManager manager, double delta, boolean check)` Creates a new instance of this class, with default stopping criterion and custom iteration manager.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `boolean` | `getCheck()` Returns `true` if positive-definiteness should be checked for both matrix and preconditioner.
    `RealVector` | `solveInPlace(RealLinearOperator a, RealLinearOperator m, RealVector b, RealVector x0)` Returns an estimate of the solution to the linear system A · x = b.

    - ### Methods inherited from class org.apache.commons.math3.linear.PreconditionedIterativeLinearSolver

      `checkParameters, solve, solve, solve, solve, solveInPlace`

    - ### Methods inherited from class org.apache.commons.math3.linear.IterativeLinearSolver

      `checkParameters, getIterationManager`

    - ### Methods inherited from class java.lang.Object

      `clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`

- - ### Field Detail

    - #### OPERATOR

          public static final String OPERATOR

      Key for the exception context.
      See Also:
        Constant Field Values

    - #### VECTOR

          public static final String VECTOR

      Key for the exception context.
      See Also:
        Constant Field Values

  - ### Constructor Detail

    - #### ConjugateGradient

          public ConjugateGradient(int maxIterations,
                           double delta,
                           boolean check)

      Creates a new instance of this class, with default stopping criterion.
      Parameters:
        `maxIterations` - the maximum number of iterations
        `delta` - the δ parameter for the default stopping criterion
        `check` - `true` if positive definiteness of both matrix and preconditioner should be checked

    - #### ConjugateGradient

          public ConjugateGradient(IterationManager manager,
                           double delta,
                           boolean check)
                            throws NullArgumentException

      Creates a new instance of this class, with default stopping criterion and custom iteration manager.
      Parameters:
        `manager` - the custom iteration manager
        `delta` - the δ parameter for the default stopping criterion
        `check` - `true` if positive definiteness of both matrix and preconditioner should be checked
      Throws:
        `NullArgumentException` - if `manager` is `null`

  - ### Method Detail

    - #### getCheck

          public final boolean getCheck()

      Returns `true` if positive-definiteness should be checked for both matrix and preconditioner.
      Returns:
        `true` if the tests are to be performed

    - #### solveInPlace

          public RealVector solveInPlace(RealLinearOperator a,
                                RealLinearOperator m,
                                RealVector b,
                                RealVector x0)
                                  throws NullArgumentException,
                                         NonPositiveDefiniteOperatorException,
                                         NonSquareOperatorException,
                                         DimensionMismatchException,
                                         MaxCountExceededException

      Returns an estimate of the solution to the linear system A · x = b. The solution is computed in-place (initial guess is modified).
      Specified by:
        `solveInPlace` in class `PreconditionedIterativeLinearSolver`
      Parameters:
        `a` - the linear operator A of the system
        `m` - the preconditioner, M (can be `null`)
        `b` - the right-hand side vector
        `x0` - the initial guess of the solution
      Returns:
        a reference to `x0` (shallow copy) updated with the solution
      Throws:
        `NonPositiveDefiniteOperatorException` - if `a` or `m` is not positive definite
        `NullArgumentException` - if one of the parameters is `null`
        `NonSquareOperatorException` - if `a` or `m` is not square
        `DimensionMismatchException` - if `m`, `b` or `x0` have dimensions inconsistent with `a`
        `MaxCountExceededException` - at exhaustion of the iteration count, unless a custom `callback` has been set at construction of the `IterationManager`

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