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

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[edit]

  1. Choose initial guess x_0\,, two other vectors x_0^* and b^*\, and a preconditioner M\,
  2. r_0 \leftarrow b-A\, x_0\,
  3. r_0^* \leftarrow b^*-x_0^*\, A^T
  4. p_0 \leftarrow M^{-1} r_0\,
  5. p_0^* \leftarrow r_0^*M^{-1}\,
  6. for k=0, 1, \ldots do
    1. \alpha_k \leftarrow {r_k^* M^{-1} r_k \over p_k^* A p_k}\,
    2. x_{k+1} \leftarrow x_k + \alpha_k \cdot p_k\,
    3. x_{k+1}^* \leftarrow x_k^* + \overline{\alpha_k}\cdot p_k^*\,
    4. r_{k+1} \leftarrow r_k - \alpha_k \cdot A p_k\,
    5. r_{k+1}^* \leftarrow r_k^*- \overline{\alpha_k} \cdot p_k^*\, A
    6. \beta_k \leftarrow {r_{k+1}^* M^{-1} r_{k+1} \over r_k^* M^{-1} r_k}\,
    7. p_{k+1} \leftarrow M^{-1} r_{k+1} + \beta_k \cdot p_k\,
    8. p_{k+1}^* \leftarrow r_{k+1}^*M^{-1}  + \overline{\beta_k}\cdot 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 \overline{\alpha} is the complex conjugate.

Unpreconditioned version of the algorithm[edit]

  1. Choose initial guess x_0\,,
  2. r_0 \leftarrow b-A\, x_0\,
  3. \hat{r}_0 \leftarrow \hat{b} - \hat{x}_0A^T
  4. p_0 \leftarrow r_0\,
  5. \hat{p}_0 \leftarrow \hat{r}_0\,
  6. for k=0, 1, \ldots do
    1. \alpha_k \leftarrow {\hat{r}_k r_k \over \hat{p}_k A p_k}\,
    2. x_{k+1} \leftarrow x_k + \alpha_k \cdot p_k\,
    3. \hat{x}_{k+1} \leftarrow \hat{x}_k + \alpha_k \cdot \hat{p}_k\,
    4. r_{k+1} \leftarrow r_k - \alpha_k \cdot A p_k\,
    5. \hat{r}_{k+1} \leftarrow \hat{r}_k- \alpha_k \cdot \hat{p}_k A^T
    6. \beta_k \leftarrow {\hat{r}_{k+1} r_{k+1} \over \hat{r}_k r_k}\,
    7. p_{k+1} \leftarrow r_{k+1} + \beta_k \cdot p_k\,
    8. \hat{p}_{k+1} \leftarrow \hat{r}_{k+1}  + \beta_k \cdot \hat{p}_k\,

Discussion[edit]

The biconjugate gradient method is numerically unstable[citation needed] (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}\left(b - A x_j \right),
x_k^*:= x_j^*+\left(b^*- x_j^* A \right) P_k A^{-1},

where j<k using the related projection

P_k:= \mathbf{u}_k \left(\mathbf{v}_k^* A \mathbf{u}_k \right)^{-1} \mathbf{v}_k^* A,

with

\mathbf{u}_k=\left[u_0, u_1, \dots, u_{k-1} \right],
\mathbf{v}_k=\left[v_0, v_1, \dots, v_{k-1} \right].

These related projections may be iterated themselves as

P_{k+1}= P_k+ \left( 1-P_k\right) u_k \otimes {v_k^* A\left(1-P_k \right) \over v_k^* A\left(1-P_k \right) 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}\left(A x_k -b \right), where

A_{k+1}^{-1}= A_k^{-1}+ \left( 1-A_k^{-1}A\right) u_k \otimes {v_k^* \left(1-A A_k^{-1} \right) \over v_k^* A\left(1-A_k^{-1}A \right) u_k}.

The new directions

p_k = \left(1-P_k \right) u_k,
p_k^* = v_k^* A \left(1- P_k \right) 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 \left( 1- P_k \right) A^{-1} r_j,
r_k^*= r_j^* \left( 1- P_k \right)

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[edit]

  • The sequences produced by the algorithm are biorthogonal, i.e., p_i^*Ap_j=r_i^*M^{-1}r_j=0 for i \neq j.
  • if P_{j'}\, is a polynomial with \mathrm{deg}\left(P_{j'}\right)+j<k, then r_k^*P_{j'}\left(M^{-1}A\right)u_j=0. The algorithm thus produces projections onto the Krylov subspace.
  • if P_{i'}\, is a polynomial with i+\mathrm{deg}\left(P_{i'}\right)<k, then v_i^*P_{i'}\left(AM^{-1}\right)r_k=0.

See also[edit]

References[edit]