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The Insidious Nature of Instability

: Studying the spectral radius and conditions under which these methods reach a solution. Modern Krylov Subspace Methods Conjugate Gradient (CG) : Primarily for symmetric positive-definite systems. GMRES and BiCGSTAB : Methods for general non-symmetric systems. Preconditioning

• Stochastic Calculus for Finance II by Steven E. Shreve – The gold standard. Read chapters 3–6 intensively.
• Brownian Motion and Stochastic Calculus by Karatzas & Shreve – More rigorous; for the brave.

Conclusion

5. Recommended References

at Georgia Tech, which focuses on modern techniques for solving large-scale linear and nonlinear systems. Georgia Institute of Technology Course Overview

• Parameter set: a = 0.1, b = 0.9, D_u = 1, D_v = 40, L chosen so that k_c fits domain.
• Linear analysis predicts μ_c ≈ 0.0X and k_c ≈ 3.2.
• Simulations show stripes for aspect ratio 1, spots for larger D_v, and hysteresis when sweeping a through μ_c (indicates subcritical bifurcation).
• Bifurcation diagram: stable homogeneous → unstable → patterned steady states; saddle-node points observed at μ ≈ μ_sn.