Debug for SYMGS+SPMV fusion #2
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Previous implementation of SYMGS+SPMV was using x_old to calculate the spmv (L * old_x + (D+U) * new_x), It was totally wrong due to the expression is equal to the value of source (i.e., b). The mathematical proof is that the backward induction is exact (U+D)x’’+Lx’=b. Now we ensure all x are new, and we calculate Ax by (L+D+U)x = b-Lx_pre+Lx to reuse the data calculated in forward sweep.
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Previous implementation of SYMGS+SPMV was using x_old to calculate the spmv (L * old_x + (D+U) * new_x), It was totally wrong due to the expression is equal to the value of source (i.e., b). The mathematical proof is that the backward induction is exact (U+D)x’’+Lx’=b.
It can also be proved by the applied code: ComputeSYMGS.cpp:764-768
totalContribution += xv[row] * currentDiagonal; while xv[row]=（rv[row] - totalContribution）/currentDiagonal, which gives totalContribution =rv[row]
Now we ensure all x are new, and we calculate Ax by (L+D+U)x = b-Lx_pre+Lx to reuse the data calculated in forward sweep.