Final answer:
To compute the QR factorization of A without multiplying matrices A₁, ..., Aₚ, sequential QR factorization for each Aᵢ is performed, thus preventing explicit multiplication of the entire sequence.
Step-by-step explanation:
To compute the QR factorization of the product A = Ap ... A2A1 without explicitly multiplying the matrices A1, ..., Ap together, we can perform the QR factorization sequentially on each matrix. Since each Ai is assumed to be square and nonsingular, we can compute its QR factorization Ai = QiRi. Starting with A1, we get its QR as A1 = Q1R1. Then, we move onto A2 and factorize the product R1A2.
Continuing this process, for each i from 2 to p, we compute the QR of the product Ri-1Ai, adjusting the previous Ri-1, which will lead to the overall QR factorization of the product A without needing to multiply all Ai matrices explicitly. This process is sometimes referred to as the sequential QR factorization or the product QR algorithm.
Note that the final Q in QA = QR is the product of all Qis, and R is the final Rp resulting from the last factorization performed.