Use multiple dispatch for verbose processing to eliminate runtime dispatches

Replaced if-else type checking with multiple dispatch methods for
processing the verbose parameter. This eliminates runtime dispatches
that JET was detecting.

Changes:
- Created _process_verbose_param helper methods using multiple dispatch
- One method for each supported type: LinearVerbosity, AbstractVerbosityPreset, Bool
- The verbose processing now compiles to direct method dispatch
- Fully type-stable for all supported verbose parameter types

This should eliminate the runtime dispatches seen in JET analysis.

Author: ChrisRackauckas

src/common.jl

@@ -232,6 +232,18 @@ default_alias_b(::AbstractSparseFactorization, ::Any, ::Any) = true
 
 DEFAULT_PRECS(A, p) = IdentityOperator(size(A)[1]), IdentityOperator(size(A)[2])
 
+# Helper functions for processing verbose parameter with multiple dispatch (type-stable)
+@inline _process_verbose_param(verbose::LinearVerbosity) = (verbose, verbose)
+@inline function _process_verbose_param(verbose::SciMLLogging.AbstractVerbosityPreset)
+    verbose_spec = LinearVerbosity(verbose)
+    return (verbose_spec, verbose_spec)
+end
+@inline function _process_verbose_param(verbose::Bool)
+    # @warn "Using `true` or `false` for `verbose` is being deprecated."
+    verbose_spec = verbose ? LinearVerbosity() : LinearVerbosity(SciMLLogging.None())
+    return (verbose_spec, verbose)
+end
+
 """
     __init_u0_from_Ab(A, b)
 
@@ -324,26 +336,7 @@ function __init(prob::LinearProblem, alg::SciMLLinearSolveAlgorithm,
         copy(A)
     end
 
-    if verbose isa LinearVerbosity
-        verbose_spec = verbose
-        init_cache_verb = verbose_spec
-    elseif verbose isa SciMLLogging.AbstractVerbosityPreset
-        verbose_spec = LinearVerbosity(verbose)
-        init_cache_verb = verbose_spec
-    elseif verbose isa Bool
-        # @warn "Using `true` or `false` for `verbose` is being deprecated. Please use a `LinearVerbosity` type to specify verbosity settings.
-        # For details see the verbosity section of the common solver options documentation page."
-        init_cache_verb = verbose
-        if verbose
-            verbose_spec = LinearVerbosity()
-        else
-            verbose_spec = LinearVerbosity(SciMLLogging.None())
-        end
-    else
-        # Fallback for any other type
-        verbose_spec = verbose
-        init_cache_verb = verbose_spec
-    end
+    verbose_spec, init_cache_verb = _process_verbose_param(verbose)
 
     b = if issparsematrix(b) && !(A isa Diagonal)
         Array(b) # the solution to a linear solve will always be dense!