### Table 4: Blocked back-substitution

in Acceleration of First and Higher Order Recurrences on Processors with Instruction Level Parallelism

1993

"... In PAGE 9: ...Blocked back-substitution techniques also work for WHILE-loops but only DO-loops will be discussed here. Using the conditioning approach described above, we parallelize the reduction recurrence applying blocking as shown in Table4 . Consider the first loop in the code.... ..."

Cited by 9

### Table 11: Code schema for blocked back-substitution

in Acceleration of First and Higher Order Recurrences on Processors with Instruction Level Parallelism

1993

"... In PAGE 15: ... 4.4 Blocked Back-substitution The schema for the code generated by the blocked back-substitution technique is given in Table11 ; see Section 2.3 for the definitions of k(n,b) and e(n, b).... ..."

Cited by 9

### Table 5: Blocked back-substitution with invariant terms

in Acceleration of First and Higher Order Recurrences on Processors with Instruction Level Parallelism

1993

Cited by 9

### Table 8: Blocked back-substituted first order multiply-add recurrence

1993

"... In PAGE 12: ...3.2 Blocked Back-substitution The schema of Table8 shows the code for the first order multiply-add recurrence after blocked back-substitution. As in section 2.... ..."

Cited by 9

### Table 12 summarizes the performance metrics for the original program and for the programs obtained after applying the symmetric and the blocked back-substitution

1993

"... In PAGE 17: ... The discussion in the context of this architecture is more appropriate for arithmetic or linear recurrences as some of the recent architectures provide fused multiply-add operation for floating-point numbers. Table12 : Performance metrics for the case of non-zero loop-variant coefficients. In the formulas, m is the order of the original recurrence,... In PAGE 18: ... First, we discuss the performance metrics in the context of the architecture in which each unit is capable of performing multiply and add operations. For the original program, the right to left order of summing terms gives the minimum value of RecMII and is used to derive the value shown in Table12 . The number of operations performed in each iteration is simply 2m.... ..."

Cited by 9

### Table E.10: Update equations for multichannel inverse modelling. By substituting the block diagonal matrices to diagonal matrices, e.g. CFCZ AX

2000

### Table 2 summarizes the performance metrics for this case. The formulas for RecMII remain the same as in the last section(see Table 1), since moving the computation of coefficients out of the loop has no impact on the RecMII of the loop. As expected, the number of operations per iterations is substantially lower than in the last section. For symmetric back-substitution, it is identical to that for the original program. For blocked back-substitution, it is identical to the original program for architectures with add and multiply capabilities but slightly higher than the original program for architectures with fused multiply-add capability. The reason for higher operation count is that log-tree summation order used in the case of blocked back-substitution is not as suited as right to left order for combining multiplications with additions.

1993

"... In PAGE 28: ... Table2 : Performance metries for the case of non-zero loop-invariant coefficients. In the formulas, m is the d is th bstituti is the l... ..."

Cited by 4

### Table 6.5: Excess entropy for substitution matrix regularizers with pseudocounts and pseudocounts plus scaled counts applied to the full blocks database. residue

1995

Cited by 21

### Table E.6: Update equations for multichannel system identification. By substituting the block diagonal matrices to diagonal matrices, e.g. C0CZ AX AM C0CZ, etc., we obtain the update

2000

### Table 4 summarizes the performance metrics for the case when all the coefficients except Cj are non-zero loop-invariant values. As pointed out in Section 4.6, the computation of coefficients in this case can be moved out of the loop. Thus, both symmetric and blocked back-substitution techniques perform about the same number of operations per iteration as the original program. The formulas for RecMII are identical to those in Table 3.

1993

"... In PAGE 30: ... Table4 : Performance metrics for the case when c, = oand the remaining coefficients have non-zero loop invariant values. In the formulas, m is the order of the original recurrence, b is the degree of back- bstituti d z is the I... ..."

Cited by 4