add verbosity documentation

Author: jClugstor

docs/src/basics/common_solver_opts.md

@@ -26,3 +26,92 @@ solve completely. Error controls only apply to iterative solvers.
   - `maxiters`: The number of iterations allowed. Defaults to `length(prob.b)`
   - `Pl,Pr`: The left and right preconditioners, respectively. For more information,
     see [the Preconditioners page](@ref prec).
+
+## Verbosity Controls
+
+The verbosity system in LinearSolve.jl provides fine-grained control over the diagnostic messages, warnings, and errors that are displayed during the solution of linear systems.
+
+The verbosity system is organized hierarchically into three main categories:
+
+1. Error Control - Messages related to fallbacks and error handling
+2. Performance - Messages related to performance considerations
+3. Numerical - Messages related to numerical solvers and iterations
+
+Each category can be configured independently, and individual settings can be adjusted to suit your needs.
+
+### Verbosity Levels
+The following verbosity levels are available:
+
+#### Individual Settings
+These settings are meant for individual settings within a category. These can also be used to set all of the individual settings in a group to the same value.
+- Verbosity.None() - Suppress all messages
+- Verbosity.Info() - Show message as log message at info level
+- Verbosity.Warn() - Show warnings (default for most settings)
+- Verbosity.Error() - Throw errors instead of warnings
+- Verbosity.Level(n) - Show messages with a log level setting of n
+
+#### Group Settings
+These settings are meant for controlling a group of settings. 
+- Verbosity.Default() - Use the default settings
+- Verbosity.All() - Show all possible messages
+
+### Basic Usage 
+
+#### Global Verbosity Control
+
+```julia 
+using LinearSolve
+
+# Suppress all messages
+verbose = LinearVerbosity(Verbosity.None())
+prob = LinearProblem(A, b)
+sol = solve(prob; verbose=verbose)
+
+# Show all messages
+verbose = LinearVerbosity(Verbosity.All())
+sol = solve(prob; verbose=verbose)
+
+# Use default settings
+verbose = LinearVerbosity(Verbosity.Default())
+sol = solve(prob; verbose=verbose)
+```
+
+#### Group Level Control 
+
+```julia 
+# Customize by category
+verbose = LinearVerbosity(
+    error_control = Verbosity.Warn(),   # Show warnings for error control related issues
+    performance = Verbosity.None(),     # Suppress performance messages
+    numerical = Verbosity.Info()        # Show all numerical related log messages at info level
+)
+
+sol = solve(prob; verbose=verbose)
+```
+
+#### Fine-grained Control
+The constructor for `LinearVerbosity` allows you to set verbosity for each specific message toggle, giving you fine-grained control. 
+The verbosity settings for the toggles are automatically passed to the group objects. 
+```julia
+# Set specific message types
+verbose = LinearVerbosity(
+    default_lu_fallback = Verbosity.Info(),                     # Show info when LU fallback is used
+    KrylovJL_verbosity = Verbosity.Warn(),                      # Show warnings from KrylovJL
+    no_right_preconditioning = Verbosity.None(),                # Suppress right preconditioning messages
+    KrylovKit_verbosity = Verbosity.Level(KrylovKit.WARN_LEVEL) # Set KrylovKit verbosity level using KrylovKit's own verbosity levels
+)
+
+sol = solve(prob; verbose=verbose)
+
+```
+
+#### Verbosity Levels
+##### Error Control Settings
+- default_lu_fallback: Controls messages when falling back to LU factorization (default: Warn)
+##### Performance Settings
+- no_right_preconditioning: Controls messages when right preconditioning is not used (default: Warn)
+##### Numerical Settings
+- using_IterativeSolvers: Controls messages when using the IterativeSolvers.jl package (default: Warn)
+- IterativeSolvers_iterations: Controls messages about iteration counts from IterativeSolvers.jl (default: Warn)
+- KrylovKit_verbosity: Controls messages from the KrylovKit.jl package (default: Warn)
+- KrylovJL_verbosity: Controls verbosity of the KrylovJL.jl package (default: None)