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Biconjugate gradient method

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In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations

Contents

A x = b .

Unlike the conjugate gradient method, this algorithm does not require the matrix A to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose A*.

The algorithm

  1. Choose initial guess x 0 , two other vectors x 0 ∗ and b ∗ and a preconditioner M
  2. r 0 ← b − A x 0
  3. r 0 ∗ ← b ∗ − x 0 ∗ A T
  4. p 0 ← M − 1 r 0
  5. p 0 ∗ ← r 0 ∗ M − 1
  6. for k = 0 , 1 , … do
    1. α k ← r k ∗ M − 1 r k p k ∗ A p k
    2. x k + 1 ← x k + α k ⋅ p k
    3. x k + 1 ∗ ← x k ∗ + α k ¯ ⋅ p k ∗
    4. r k + 1 ← r k − α k ⋅ A p k
    5. r k + 1 ∗ ← r k ∗ − α k ¯ ⋅ p k ∗ A
    6. β k ← r k + 1 ∗ M − 1 r k + 1 r k ∗ M − 1 r k
    7. p k + 1 ← M − 1 r k + 1 + β k ⋅ p k
    8. p k + 1 ∗ ← r k + 1 ∗ M − 1 + β k ¯ ⋅ p k ∗

In the above formulation, the computed r k and r k ∗ satisfy

r k = b − A x k , r k ∗ = b ∗ − x k ∗ A

and thus are the respective residuals corresponding to x k and x k ∗ , as approximate solutions to the systems

A x = b , x ∗ A = b ∗ ;

x ∗ is the adjoint, and α ¯ is the complex conjugate.

Unpreconditioned version of the algorithm

  1. Choose initial guess x 0 ,
  2. r 0 ← b − A x 0
  3. r ^ 0 ← b ^ − x ^ 0 A T
  4. p 0 ← r 0
  5. p ^ 0 ← r ^ 0
  6. for k = 0 , 1 , … do
    1. α k ← r ^ k r k p ^ k A p k
    2. x k + 1 ← x k + α k ⋅ p k
    3. x ^ k + 1 ← x ^ k + α k ⋅ p ^ k
    4. r k + 1 ← r k − α k ⋅ A p k
    5. r ^ k + 1 ← r ^ k − α k ⋅ p ^ k A T
    6. β k ← r ^ k + 1 r k + 1 r ^ k r k
    7. p k + 1 ← r k + 1 + β k ⋅ p k
    8. p ^ k + 1 ← r ^ k + 1 + β k ⋅ p ^ k

Discussion

The biconjugate gradient method is numerically unstable (compare to the biconjugate gradient stabilized method), but very important from a theoretical point of view. Define the iteration steps by

x k := x j + P k A − 1 ( b − A x j ) , x k ∗ := x j ∗ + ( b ∗ − x j ∗ A ) P k A − 1 ,

where j < k using the related projection

P k := u k ( v k ∗ A u k ) − 1 v k ∗ A ,

with

u k = [ u 0 , u 1 , … , u k − 1 ] , v k = [ v 0 , v 1 , … , v k − 1 ] .

These related projections may be iterated themselves as

P k + 1 = P k + ( 1 − P k ) u k ⊗ v k ∗ A ( 1 − P k ) v k ∗ A ( 1 − P k ) u k .

A relation to Quasi-Newton methods is given by P k = A k − 1 A and x k + 1 = x k − A k + 1 − 1 ( A x k − b ) , where

A k + 1 − 1 = A k − 1 + ( 1 − A k − 1 A ) u k ⊗ v k ∗ ( 1 − A A k − 1 ) v k ∗ A ( 1 − A k − 1 A ) u k .

The new directions

p k = ( 1 − P k ) u k , p k ∗ = v k ∗ A ( 1 − P k ) A − 1

are then orthogonal to the residuals:

v i ∗ r k = p i ∗ r k = 0 , r k ∗ u j = r k ∗ p j = 0 ,

which themselves satisfy

r k = A ( 1 − P k ) A − 1 r j , r k ∗ = r j ∗ ( 1 − P k )

where i , j < k .

The biconjugate gradient method now makes a special choice and uses the setting

u k = M − 1 r k , v k ∗ = r k ∗ M − 1 .

With this particular choice, explicit evaluations of P k and A−1 are avoided, and the algorithm takes the form stated above.

Properties

  • If A = A ∗ is self-adjoint, x 0 ∗ = x 0 and b ∗ = b , then r k = r k ∗ , p k = p k ∗ , and the conjugate gradient method produces the same sequence x k = x k ∗ at half the computational cost.
  • The sequences produced by the algorithm are biorthogonal, i.e., p i ∗ A p j = r i ∗ M − 1 r j = 0 for i ≠ j .
  • if P j ′ is a polynomial with d e g ( P j ′ ) + j < k , then r k ∗ P j ′ ( M − 1 A ) u j = 0 . The algorithm thus produces projections onto the Krylov subspace.
  • if P i ′ is a polynomial with i + d e g ( P i ′ ) < k , then v i ∗ P i ′ ( A M − 1 ) r k = 0 .

    • References

    Biconjugate gradient method Wikipedia
    (Text) CC BY-SA