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Purpose: This study aims to propose the parametric-guiding algorithm, the complex-root (CR) tunneling algorithm and the method that integrates both algorithms for the heat and fluid flow (HFF) community, and apply them to nonlinear Bratu's boundary-value problem (BVP) and Blasius BVP. Design/methodology/approach: In the first algorithm, iterations are primarily guided by a diminishing parameter that is introduced to reduce magnitudes of fictitious source terms. In the second algorithm, when iteration-related barriers are encountered, CRs are generated to tunnel through the barrier. At the exit of the tunnel, imaginary parts of CRs are trimmed. Findings: In terms of the robustness of convergence, the proposed method outperforms the traditional Newton–Raphson (NR) method. For most pulsed initial guesses that resemble pulsed initial conditions for the transient Bratu BVP, the proposed method has not failed to converge. Originality/value: To the best of the authors' knowledge, the parametric-guiding algorithm, the CR tunneling algorithm and the method that integrates both have not been reported in the computational-HFF-related literature. [ABSTRACT FROM AUTHOR]