Dear PETSc Development Team,


I am a researcher using PETSc for numerical simulations of partial differential equations. While reviewing the documentation for the SNES Picard interface, I noticed a sign inconsistency in the defect correction formulation that appears to be a typo.


The documentation opens by stating the equation to be solved is:
$$A(x)\,x = bp(x) - b$$


However, the provided defect correction iteration is written as:
$$A(x^n)(x^{n+1} - x^n) = bp(x^n) + b - A(x^n)x^n$$


Following this, the "classic" Picard iteration is given as:
$$A(x^n)x^{n+1} = bp(x^n) + b$$


There is a sign error on the constant term $b$. If we substitute the converged solution $x^*$ into the stated Picard iteration, we obtain $A(x^*)x^* = bp(x^*) + b$, which contradicts the original equation $A(x)x = bp(x) - b$ by a difference of $2b$.


The correct defect correction form consistent with the original equation should be:
$$A(x^n)(x^{n+1} - x^n) = bp(x^n) - b - A(x^n)x^n$$


which reduces to the correct classic Picard iteration:
$$A(x^n)x^{n+1} = bp(x^n) - b$$


I greatly appreciate the high-quality documentation and the extensive work your team has put into PETSc. I wanted to report this small issue to help improve the manual. Please let me know if I have misunderstood the formulation.

Best regards,

Jingdu Li