Fix the docs

Author: haampie

src/bicgstabl.jl

@@ -159,8 +159,8 @@ For BiCGStab(l) this is a less dubious term than "number of iterations";
 - `abstol::Real = zero(real(eltype(b)))`,
   `reltol::Real = sqrt(eps(real(eltype(b))))`: absolute and relative
   tolerance for the stopping condition
-  `|r_k| / |r_0| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
-  is the residual in the `k`th iteration;
+  `|r_k| ≤ max(reltol * resnorm, abstol)`, where `r_k ≈ A * x_k - b`
+  is the approximate residual in the `k`th iteration;
   !!! note
       1. The true residual norm is never computed during the iterations,
          only an approximation;

src/cg.jl

@@ -181,8 +181,11 @@ cg(A, b; kwargs...) = cg!(zerox(A, b), A, b; initially_zero = true, kwargs...)
 - `abstol::Real = zero(real(eltype(b)))`,
   `reltol::Real = sqrt(eps(real(eltype(b))))`: absolute and relative
   tolerance for the stopping condition
-  `|r_k| / |r_0| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
-  is the residual in the `k`th iteration;
+  `|r_k| ≤ max(reltol * resnorm, abstol)`, where `r_k ≈ A * x_k - b`
+  is approximately the residual in the `k`th iteration.
+  !!! note
+      The true residual norm is never explicitly computed during the iterations
+      for performance reasons; it may have accumulate rounding errors.
 - `maxiter::Int = size(A,2)`: maximum number of iterations;
 - `verbose::Bool = false`: print method information;
 - `log::Bool = false`: keep track of the residual norm in each iteration.

src/chebyshev.jl

@@ -120,7 +120,7 @@ Solve Ax = b for symmetric, definite matrices A using Chebyshev iteration.
 - `abstol::Real = zero(real(eltype(b)))`,
   `reltol::Real = sqrt(eps(real(eltype(b))))`: absolute and relative
   tolerance for the stopping condition
-  `|r_k| / |r_0| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
+  `|r_k| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
   is the residual in the `k`th iteration;
 - `maxiter::Int = size(A, 2)`: maximum number of inner iterations of GMRES;
 - `Pl = Identity()`: left preconditioner;

src/gmres.jl

@@ -156,7 +156,7 @@ Solves the problem ``Ax = b`` with restarted GMRES.
 - `abstol::Real = zero(real(eltype(b)))`,
   `reltol::Real = sqrt(eps(real(eltype(b))))`: absolute and relative
   tolerance for the stopping condition
-  `|r_k| / |r_0| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
+  `|r_k| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
 - `restart::Int = min(20, size(A, 2))`: restarts GMRES after specified number of iterations;
 - `maxiter::Int = size(A, 2)`: maximum number of inner iterations of GMRES;
 - `Pl`: left preconditioner;

src/idrs.jl

@@ -27,7 +27,7 @@ shadow space.
 - `abstol::Real = zero(real(eltype(b)))`,
   `reltol::Real = sqrt(eps(real(eltype(b))))`: absolute and relative
   tolerance for the stopping condition
-  `|r_k| / |r_0| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
+  `|r_k| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
   is the residual in the `k`th iteration;
 - `maxiter::Int = size(A, 2)`: maximum number of iterations;
 - `log::Bool`: keep track of the residual norm in each iteration;

src/minres.jl

@@ -178,7 +178,7 @@ Solve Ax = b for (skew-)Hermitian matrices A using MINRES.
 - `abstol::Real = zero(real(eltype(b)))`,
   `reltol::Real = sqrt(eps(real(eltype(b))))`: absolute and relative
   tolerance for the stopping condition
-  `|r_k| / |r_0| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
+  `|r_k| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
   is the residual in the `k`th iteration
   !!! note
       The residual is computed only approximately.

src/qmr.jl

@@ -240,7 +240,7 @@ Solves the problem ``Ax = b`` with the Quasi-Minimal Residual (QMR) method.
 - `abstol::Real = zero(real(eltype(b)))`,
   `reltol::Real = sqrt(eps(real(eltype(b))))`: absolute and relative
   tolerance for the stopping condition
-  `|r_k| / |r_0| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
+  `|r_k| ≤ max(reltol * resnorm, abstol)`, where `r_k = A * x_k - b`
 - `log::Bool`: keep track of the residual norm in each iteration;
 - `verbose::Bool`: print convergence information during the iteration.