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  "body": "\n## Statement\n\nLet $A$ be a real [[invertible|0EB9N0S33C]] $n \\times n$ matrix, let $b \\in \\R^n$, and let $x_0 \\in \\R^n$ with $r_0 = b - A x_0$.\nChoose $\\hat{r}_0 \\in \\R^n$ with $\\hat{r}_0^T r_0 \\ne 0$, and set $p_0 = r_0$.\nThe *biconjugate gradient stabilized method* (BiCGSTAB) computes, for $k = 0, 1, \\dots$,\n$$v_k = A p_k, \\quad \\alpha_k = \\frac{\\hat{r}_0^T r_k}{\\hat{r}_0^T v_k}, \\quad s_k = r_k - \\alpha_k v_k, \\quad t_k = A s_k, \\quad \\omega_k = \\frac{t_k^T s_k}{t_k^T t_k},$$\n$$x_{k+1} = x_k + \\alpha_k p_k + \\omega_k s_k, \\quad r_{k+1} = s_k - \\omega_k t_k,$$\n$$\\beta_k = \\frac{\\hat{r}_0^T r_{k+1}}{\\hat{r}_0^T r_k} \\cdot \\frac{\\alpha_k}{\\omega_k}, \\quad p_{k+1} = r_{k+1} + \\beta_k (p_k - \\omega_k v_k).$$\nSuppose that all denominators are nonzero up to step $m$.\nThen in exact arithmetic, for $k \\le m$:\n\n1. $r_k = b - A x_k$, and $r_k = Q_k(A) P_k(A) r_0$, where $P_k$ is the residual polynomial of the [[biconjugate gradient method|HGF11AHX5R]] with shadow residual $\\hat{r}_0$, and $Q_k(t) = \\prod_{j=0}^{k-1} (1 - \\omega_j t)$;\n2. each $\\omega_k$ minimizes $\\norm{s_k - \\omega t_k}_2$ over $\\omega \\in \\R$;\n3. each step uses two products with $A$ and none with $A^T$.\n\n## Remarks\n\nThe residual polynomial $P_k$ of BiCG has degree $k$ and value $1$ at $0$, so $r_k$ comes from a polynomial of degree $2k$ in the [[Krylov subspace|QB45PCGQPV]] $\\mathcal{K}_{2k+1}(A, r_0)$.\nThe recurrences come from writing the BiCG coefficients in terms of the inner products $\\hat{r}_0^T Q_k(A) P_k(A) r_0$, which needs no $A^T$ because $\\hat{r}_0^T Q_k(A) = (Q_k(A^T) \\hat{r}_0)^T$.\nThe factor $Q_k$ is a product of local one-step minimal residual steps, which smooths the erratic convergence of BiCG and of the method CGS.\nExample: if $r_0$ is an eigenvector of $A$ for $\\lambda$, then $v_0 = \\lambda r_0$, $\\alpha_0 = 1/\\lambda$ and $s_0 = 0$, so $x_0 + \\alpha_0 p_0$ already solves the system.\nIn that case $t_0 = 0$ and $\\omega_0$ is undefined, and an implementation stops when $s_k$ is small.\nThe method can still break down, when $\\hat{r}_0^T r_k = 0$ or $\\omega_k = 0$.\nIt can stagnate when the $\\omega_k$ are close to $0$, which tends to happen when $A$ has eigenvalues with large imaginary parts.\n\n## Explanations\n\n### L1 human\n\nBiCGSTAB is a fix of the biconjugate gradient method for nonsymmetric systems.\nIt avoids the transpose of the matrix, and after each BiCG-like step it adds a small step that reduces the residual as much as possible.\nThe result usually converges more smoothly than BiCG and costs little memory.\n\n### L2 human\n\nBiCGSTAB, due to van der Vorst, is with restarted [[GMRES|A3FR6PNEPB]] one of the default Krylov [[iterative methods|2RPE77ACER]] for nonsymmetric sparse systems.\nIts memory and work per step are fixed, in contrast with full GMRES, but it has no global minimization property.\nIt is used with a [[preconditioner|KE98SW3SBD]], typically an incomplete LU factorization.\nVariants such as BiCGSTAB($\\ell$) use minimal residual polynomials of higher degree to handle complex eigenvalues better.\n",
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