Abstract:
A loop unrolling trasformation specified by loop unrolling factors UF[1], . . . , UF[k] is performed on a perfect nest of k multiple loops to produce an unrolled loop representation as follows. Moving from the outermost loop to the innermost loop of the nest, the unroll factor UF[j] of the current loop is examined. First, the separate unrolled loop body is expanded by the specified unroll factor UF[j]. Second, the loop header for the current loop is adjusted so that if the loop&#39;s iteration count, COUNT[j], is known to be less than or equal to the unroll factor, UP[j], then the loop header is simply an assignment of the index variable to the lower-bound expression; otherwise, the loop header is adjusted so that the unrolled loop&#39;s iteration count equals .left brkt-bot.COUNT[J]/UF[J].right brkt-bot. a rounded down truncation of the division. Third, a remainder loop nest is generated, if needed. The size of the generated code when unrolling multiple nested loops is substantially reduced. The proportion of the object code comprising lower execution frequency remainder loops is also substantially reduced. The compile-time of unrolled multiple nested loops is also substantially reduced.

Description:
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     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates in general to software optimization that may be used by a programmer or an automatic program optimizer, and more particularly to a technique for generating compact code for the unrolling transformation applied to a perfect nest of loops. 
     2. Description of the Related Art 
     Loop unrolling is a well known program transformation used by programmers and program optimizers to improve the instruction-level parallelism and register locality and to decrease the branching overhead of program loops. These benefits arise because creating multiple copies of the loop body provides more opportunities for program optimization. Most optimizing compilers employ the loop unrolling transformation to some degree. In addition, many software packages, especially those for matrix computations, contain library routines in which loops have been hand-unrolled for improved performance. 
     Historically, the unrolling transformation has been defined for a single loop. More recently, it has been observed that further benefits may be obtained by unrolling multiple nested loops and fusing or jamming together unrolled copies of inner loops. This &#34;unroll-and jam&#34; transformation has a multiplicative effect on the unroll factors of the individual loops, such that the number of copies in the unrolled loop body equals the product of the unroll factors of the individual loops. 
     A general concern with the aggressive use of unrolling is with the size of the code generated after the loop unrolling transformation is performed, especially when unrolling multiple perfectly nested loops. Apart from creating a larger unrolled loop body, additional loops have to be introduced to correctly handle cases where the unroll factor does not evenly divide the number of iterations. These &#34;remainder&#34; loops substantially increase the compile-time for the transformed code and the size of the final object code, even though only a small fraction of the program&#39;s execution time is spent in these remainder loops. 
     To quantitatively understand the substantial code size increase that may result from applying the unroll-and-jam transformation to multiple loops, let UF[1], . . . , UF[k] be the unroll factors for a perfect nest of k loops numbered from outermost to innermost. After the conventional unroll-and-jam transformation is applied, the unrolled loop will contain |UF[1]+mod(1,UF[1])|×|UF[2]+mod(1,UF[2]).vertline.× . . . ×|UF[k]+mod(1,UF[k])|copies of the loop body for the general and common case of loop bounds that are unknown at compile-time, where &#34;mod(1, UF[i])&#34; is a function that equals zero when UF[i]=1 and where the function equals one when UF[i]&gt;1. For example, if all k loops have the same unroll factor m where UF[1]=UF[2]= . . . =UF[k]=m&gt;1, then the conventional unroll-and-jam transformation will generate C&#39;=(m+1) k  copies of the loop body in the unrolled code. More specifically, if the conventional unroll-and-jam transformation is applied to the example code of Table A with m=4 and k=3, then 125 copies will be generated, C&#39;=125=(4+1) 3 . This generated code is shown in Table C. Note that only 64 copies are in the main unrolled loop body, and that the remaining 65 copies are lower execution frequency remainder loops. Thus, this example clearly illustrates the high proportion of code comprising lower execution frequency remainder loops, as in this example, the prior art generates a greater amount of code in lower execution frequency remainder loops than it does in the higher execution frequency main unrolled loop body. 
     Thus, conventional techniques of loop unrolling in generating remainder loops substantially increase the compile-time, the size of the object code, and the proportion of the object code comprising lower execution frequency remainder loops. As such, there is a need for a method of, and apparatus for, and article of manufacture for generating smaller more compact code for the unrolling transformation of multiple nested loops. 
     SUMMARY OF THE INVENTION 
     The invention disclosed herein comprises a method of, a system for, and an article of manufacture for generating compact code for the unrolling transformation. The present invention substantially reduces the size of the generated code when unrolling multiple nested loops. The present invention substantially reduces the proportion of the object code comprising lower frequency execution remainder loops. The present invention substantially reduces the compile-time of unrolled multiple nested loops. 
     Given a perfect nest of k multiple loops, the present invention performs a loop unrolling transformation specified by loop unrolling factors UF[1], . . . , UF[k] and produces an unrolled loop representation as follows. Moving from the outermost loop to the innermost loop of the nest, the invention examines the unroll factor UF[j] of the current loop. The first major step is to expand the separate unrolled loop body by the specified unroll factor UF[j]. The second major step is to adjust the loop header for the current loop. If the loop&#39;s iteration count, COUNT[j], is known to be less than or equal to the unroll factor, UF[j], then the loop header is simply an assignment of the index variable to the lower-bound expression; otherwise, the loop header is adjusted so that the unrolled loop&#39;s iteration count equals .left brkt-bot.COUNT[j]/UF[j].right brkt-bot., a rounded down truncation of the division. The third major step is to generate a remainder loop nest, if needed. The body of the remainder loop nest is a single copy of the input loop body. The body of the remainder loop is a cross product of unrolled copies from loops 1 . . . j-1 and single copies from loops j . . . k. The remainder loop is not created if it is determined at compile time that the loop length COUNT[j] is a multiple of the unroll factor UF[j]. 
     As opposed to the conventional unroll-and-jam transformation discussed above which produces |UF[1]+mod(1,UF[1])|×|UF[2]+mod(1,UF[2]).vertline. . . . ×|UF[k]+mod(1,UF[k])| loop copies in the transformed code, the present invention only produces UF[1]× . . . ×UF[k]copies of the code from the original loop body in addition to at most (UF[1]× . . . ×UF[k-1])+(UF[1]× . . . ×UF[k-2])+ . . . +(UF[1]×UF[2])+(UF[1])+1 remainder loops in the transformed code, each containing a single copy of the code from the original loop body. More precisely, the number of remainder loops generated by the present invention is at most: SUM FROM i=1 to k(PRODUCT FROM j=1 to k-i (UF[j])). For the k=1 single loop case, the present invention produces the same number of copies as the conventional unroll-and-jam transformation, at most UF[1]+1 copies: UF[1] copies of the unrolled loop and at most one remainder loop. However, for the k≧2 multiple-loop case, the present invention generates substantially fewer lower execution frequency remainder loops than the conventional unroll-and-jam transformation. 
     Referring back to the example code of Table A where application of the conventional unroll-and-jam transformation produced 125 copies of the loop body, if the unroll transformation of the present invention is applied to the example code of Table A with m=4 and k=3, then only 85 copies will be generated, C=1+m+m 2  + . . . +m k  =|m.sup.(k+ 1-1|/|m-1|=|4.sup.(3+1) -1|/|4-1|=85. The transformed code generated by the present invention for the example code is shown in Table B. Note that only 21 copies (85 total-64 main unrolled loop body copies) of the lower execution frequency remainder loops are generated by the present invention as opposed to the 61 copies of lower execution frequency remainder loops generated by the prior art. 
     The present invention has the advantage of generating fewer remainder loop statements after unrolling, and hence reducing the compilation time to process the output generated by the unrolling transformation. 
     The present invention has the further advantage of reducing the code size that results after the unrolling transformation is performed. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     For a more complete understanding of the present invention and the advantages thereof, reference is now made to the Detailed Description in conjunction with the attached Drawings, in which: 
     FIG. 1 illustrates a symbolic pictorial representation of a partitioning of loop iterations for the simple case of k=1 (a perfect nest of a single loop) in accordance with the present invention; 
     FIG. 2 illustrates a symbolic pictorial representation of a partitioning of loop iterations for the simple case of k=2 (a perfect nest of two loops) in accordance with the present invention; 
     FIG. 3 illustrates psuedo-code comprising the operations preferred in carrying out the present invention; 
     FIG. 4 is a flowchart illustrating the operations preferred in carrying out the present invention; 
     FIG. 5 illustrates a tree structure for an original intermediate representation of loops to be unrolled in accordance with the present invention; 
     FIG. 6 and FIG. 7 illustrate unrolled loop bodies produced in accordance with the present invention; 
     FIG. 8, FIG. 9, and FIG. 10 are additional flowcharts illustrating the operations preferred in carrying out the present invention; and 
     FIG. 11 is a block diagram of a computer system used in performing the method of the present invention, forming part of the apparatus of the present invention, and which may use the article of manufacture comprising a computer-readable storage medium having a computer program embodied in said medium which may cause the computer system to practice the present invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Referring now to FIG. 1 and FIG. 2, pictorial representations of partitioning of loop iterations according to the present invention are shown for the simple cases of k=1 (a perfect nest of a single loop) and k=2 (a perfect nest of two loops). FIG. 1 illustrates that application of the present invention to a perfect nest of a single loop (k=1) produces UF[1]+1 copies of the unrolled loop body, UF[1] copies in the unrolled loop and one copy in the remainder loop. FIG. 2 illustrates that application of the present invention to a perfect nest of two loops (k=2) produces UF[1]×UF[2]+UF[1]+1 copies of the unrolled loop body, UF[1]×UF[2] copies in the unrolled loop and UF[1]+1 copies in the remainder loop. 
     Referring now to Table A, there is shown an example comprising a perfect nest of four loops, the outer three of which are each to be unrolled with an unroll factor of four (UF[1]=UF[2]=UF[3]=4). Thus, the innermost loop is considered a part of the unrolled loop body as the innermost loop is not unrolled in this example. 
     
                       TABLE A______________________________________do l = 1, ndo k = 1, ndo j = 1, ndo i = 1, nsum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)end doend doend doend do______________________________________ 
    
     Applying the unroll transformation of the present invention to the perfect nest of loops of Table A with unroll factors UF[1]=UF[2]=UF[3]=4 produces the unrolled loop of Table B containing 64 copies of the code from the original loop body (UF[1]× . . . × UF[3]=4×4×4=64) and also containing 21 remainder loops, each containing a single copy of the code from the original loop body ((UF[1]× . . . ×UF[k-1])+(UF[1]× . . . ×UF[k-2])+ . . . +(UF[1]×UF[2])+(UF[1])+1=(UF[1]×UF[2])+(UF[1])+1=16+4+1=21). 
     To understand how the loop unrolling of the present invention can improve register locality, notice that the original loop in Table A has no register locality for arrays a, b, and c as the original loop performs three load operations per iteration. However, the main unrolled loop body in Table B has an abundant amount of register locality, e.g., the value of a(i,j,k) can be loaded once and reused in a register in the first four statements of the unrolled loop body. In fact, of the 64*3=192 references to arrays a, b, and c in the main unrolled loop body, only 48 contain distinct values. If the target processor has sufficient registers to hold these 48 array elements, then the main unrolled loop would only perform 48 load operations. However, the original loop would have performed 192 loads for 64 iterations. Thus, the practice of the present invention reduces the number of load operations performed by a factor of four in this example. 
     If the program semantics permit reassociation of addition (+) operations, loop unrolling can also help reveal more instruction-level parallelism in this example. For example, a partial sum can be computed for the first 32 statements on one functional unit, another partial sum can be computed in parallel for the remaining 32 statements on another functional unit, and the two partial sums could be combined by a single add operation at the end. 
     
                       TABLE B______________________________________do l=1,n - 3,4 do k=1,n - 3,4 do j=1,n - 3,4  do i=1,n,1  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  sum = sum + a(i,j,k) + b(i,j,l + 1) + c(i,k,l + 1)  sum = sum + a(i,j,k) + b(i,j,l + 2) + c(i,k,l + 2)  sum = sum + a(i,j,k) + b(i,j,l + 3) + c(i,k,l + 3)  sum = sum + a(i,j,k + 1) + b(i,j,l) + c(i,k + 1,l)  sum = sum + a(i,j,k + 1) + b(i,j,l + 1) + c(i,k + 1,l + 1)  sum = sum + a(i,j,k + 1) + b(i,j,l + 2) + c(i,k + 1,l + 2)  sum = sum + a(i,j,k + 1) + b(i,j,l + 3) + c(i,k + 1,l + 3)  sum = sum + a(i,j,k + 2) + b(i,j,l) + c(i,k + 2,l)  sum = sum + a(i,j,k + 2) + b(i,j,l + 1) + c(i,k + 2,l + 1)  sum = sum + a(i,j,k + 2) + b(i,j,l + 2) + c(i,k + 2,l + 2)  sum = sum + a(i,j,k + 2) + b(i,j,l + 3) + c(i,k + 2,l + 3)  sum = sum + a(i,j,k + 3) + b(i,j,l) + c(i,k + 3,l)  sum = sum + a(i,j,k + 3) + b(i,j,l + 1) + c(i,k + 3,l + 1)  sum = sum + a(i,j,k + 3) + b(i,j,l + 2) + c(i,k + 3,l + 2)  sum = sum + a(i,j,k + 3) + b(i,j,l + 3) + c(i,k + 3,l + 3)  sum = sum + a(i,j + 1,k) + b(i,j + 1,l) + c(i,k,l)  sum = sum + a(i,j + 1,k) + b(i,j + 1,l + 1) + c(i,k,l + 1)  sum = sum + a(i,j + 1,k) + b(i,j + 1,l + 2) + c(i,k,l + 2)  sum = sum + a(i,j + 1,k) + b(i,j + 1,l + 3) + c(i,k,l + 3)  sum = sum + a(i,j + 1,k + 1) + b(i,j + 1,l) + c(i,k + 1,l)  sum = sum + a(i,j + 1,k + 1) + b(i,j + 1,l + 1) + c(i,k + i,l + 1)  sum = sum + a(i,j + 1,k + 1) + b(i,j + 1,l + 2) + c(i,k + i,l + 2)  sum = sum + a(i,j + 1,k + 1) + b(i,j + 1,l + 3) + c(i,k + i,l + 3)  sum = sum + a(i,j + 1,k + 2) + b(i,j + 1,l) + c(i,k + 2,l)  sum = sum + a(i,j + 1,k + 2) + b(i,j + 1,l + 1) + c(i,k + 2,l + 1)  sum = sum + a(i,j + 1,k + 2) + b(1,j + 1,l + 2) + c(i,k.+ 2,l + 2)  sum = sum + a(i,j + 1,k + 2) + b(i,j + 1,l + 3) + c(i,k + 2,l + 3)  sum = sum + a(i,j + 1,k + 3) + b(i,j + 1,l) + c(i,k + 3,l)  sum = sum + a(i,j + 1,k + 3) + b(i,j + 1,l + 1) + c(i,k + 3,l + 1)  sum = sum + a(i,j + 1,k + 3) + b(i,j + 1,l + 2) + c(i,k + 3,l + 2)  sum = sum + a(i,j + 1,k + 3) + b(i,j + 1,l + 3) + c(i,k + 3,l + 3)  sum = sum + a(i,j + 2,k) + b(i,j + 2,l) + c(i,k,l)  sum = sum + a(i,j + 2,k) + b(i,j + 2,l + 1) + c(i,k,l + 1)  sum = sum + a(i,j + 2,k) + b(i,j + 2,l + 2) + c(i,k,l + 2)  sum = sum + a(i,j + 2,k) + b(i,j + 2,l + 3) + c(i,k,l + 3)  sum = sum + a(i,j + 2,k + 1) + b(i,j + 2,l) + c(i,k + 1,l)  sum = sum + a(i,j + 2,k + 1) + b(i,j + 2,l + 1) + c(i,k + 1,l + 1)  sum = sum + a(i,j + 2,k + 1) + b(i,j + 2,l + 2) + c(i,k + 1,l + 2)  sum = sum + a(i,j + 2,k + 1) + b(i,j + 2,l + 3) + c(i,k + 1,l + 3)  sum = sum + a(i,j + 2,k + 2) + b(i,j + 2,l) + c(1,k + 2,l)  sum = sum + a(i,j + 2,k + 2) + b(i,j + 2,l + 1) + c(i,k + 2,l + 1)  sum = sum + a(i,j + 2,k + 2) + b(i,j + 2,l + 2) + c(i,k + 2,l + 2)  sum = sum + a(i,j + 2,k + 2) + b(i,j + 2,l + 3) + c(i,k + 2,l + 3)  sum = sum + a(i,j + 2,k + 3) + b(i,j + 2,l) + c(i,k + 3,l)  sum = sum + a(i,j + 2,k + 3) + b(i,j + 2,l + 1) + c(i,k + 3,l + 1)  sum = sum + a(i,j + 2,k + 3) + b(i,j + 2,l + 2) + c(i,k + 3,l + 2)  sum = sum + a(i,j + 2,k + 3) + b(i,j + 2,l + 3) + c(i,k + 3,l + 3)  sum = sum + a(i,j + 3,k) + b(i,j + 3,l) + c(i,k,l)  sum = sum + a(i,j + 3,k) + b(i,j + 3,l + 1) + c(i,k,l + 1)  sum = sum + a(i,j + 3,k) + b(i,j + 3,l + 2) + c(i,k,l + 2)  sum = sum + a(i,j + 3,k) + b(i,j + 3,l + 3) + c(i,k,l + 3)  sum = sum + a(i,j + 3,k + 1) + b(i,j + 3,l) + c(i,k + 1,l)  sum = sum + a(i,j + 3,k + 1) + b(i,j + 3,l + 1) + c(i,k + i,l + 1)  sum = sum + a(i,j + 3,k + 1) + b(i,j + 3,l + 2) + c(i,k + i,l + 2)  sum = sum + a(i,j + 3,k + 1) + b(i,j + 3,l + 3) + c(i,k + i,l + 3)  sum = sum + a(i,j + 3,k + 2) + b(i,j + 3,l) + c(i,k + 2,l)  sum = sum + a(i,j + 3,k + 2) + b(i,j + 3,l + 1) + c(i,k + 2,l + 1)  sum = sum + a(i,j + 3,k + 2) + b(i,j + 3,l + 2) + c(i,k + 2,l + 2)  sum = sum + a(i,j + 3,k + 2) + b(i,j + 3,l + 3) + c(i,k + 2,l + 3)  sum = sum + a(i,j + 3,k + 3) + b(i,j + 3,l) + c(i,k + 3,l)  sum = sum + a(i,j + 3,k + 3) + b(i,j + 3,l + 1) + c(i,k + 3,l + 1)  sum = sum + a(i,j + 3,k + 3) + b(i,j + 3,l + 2) + c(i,k + 3,l + 2)  sum = sum + a(i,j + 3,k + 3) + b(i,j + 3,l + 3) + c(i,k + 3,l + 3)  end do end do do j=j,n,1  do i=1,n,1  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  sum = sum + a(i,j,k) + b(i,j,l + 1) + c(i,k,l + 1)  sum = sum + a(i,j,k) + b(i,j,l + 2) + c(i,k,l + 2)  sum = sum + a(i,j,k) + b(i,j,l + 3) + c(i,k,l + 3)  sum = sum + a(i,j,k + 1) + b(i,j,l) + c(i,k + 1,l)  sum = sum + a(i,j,k + 1) + b(i,j,l + 1) + c(i,k + 1,l + 1)  sum = sum + a(i,j,k + 1) + b(i,j,l + 2) + c(i,k + 1,l + 2)  sum = sum + a(i,j,k + 1) + b(i,j,l + 3) + c(i,k + 1,l + 3)  sum = sum + a(i,j,k + 2) + b(i,j,l) + c(i,k + 2,l)  sum = sum + a(i,j,k + 2) + b(i,j,l + 1) + c(i,k + 2,l + 1)  sum = sum + a(i,j,k + 2) + b(i,j,l + 2) + c(i,k + 2,l + 2)  sum = sum + a(i,j,k + 2) + b(i,j,l + 3) + c(i,k + 2,l + 3)  sum = sum + a(i,j,k + 3) + b(i,j,l) + c(i,k + 3,l)  sum = sum + a(i,j,k + 3) + b(i,j,l + 1) + c(i,k + 3,l + 1)  sum = sum + a(i,j,k + 3) + b(i,j,l + 2) + c(i,k + 3,l + 2)  sum = sum + a(i,j,k + 3) + b(i,j,l + 3) + c(i,k + 3,l + 3)  end do end do end do do k=k,n,1 do j=1,n,1  do i=1,n,1  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  sum = sum + a(i,j,k) + b(i,j,l + 1) + c(i,k,l + 1)  sum = sum + a(i,j,k) + b(i,j,l + 2) + c(i,k,l + 2)  sum = sum + a(i,j,k) + b(i,j,l + 3) + c(i,k,l + 3)  end do end do end doend dodo i=l,n,1 do k=1,n,1 do j=1,n,1  do i=1,n,1  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  end do end do end doend do______________________________________ 
    
     The advantages and benefits of the present invention may be appreciated by comparing the 85 copies of the loop body generated by the present invention to the 125 copies generated by the conventional unroll-and-jam transformation of the prior art as shown in Table C. The advantages and benefits of the present invention may be further appreciated by comparing the 21 copies of the lower execution frequency remainder loops generated by the present invention to the 61 copies generated by the conventional unroll-and-jam transformation of the prior art as shown in Table C. Though it may be possible to perform additional loop fusion or loop jamming on some of the remainder loops in the prior art version, the loop fusion will only affect remainder loops and will not reduce the number of copies. 
     
                       TABLE C______________________________________do l = 1, n-3, 4 do k = 1, n-3, 4 do j = 1, n-3, 4  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  sum = sum + a(i,j+1,k) + b(i,j+1,l) + c(i,k,l)  sum = sum + a(i,j+2,k) + b(i,j+2,l) + c(i,k,l)  sum = sum + a(i,j+3,k) + b(i,j+3,l) + c(i,k,l)  sum = sum + a(i,j,k+1) + b(i,j,l) + c(i,k+1,l)  sum = sum + a(i,j+1,k+1) + b(i,j+1,l) + c(i,k+1,l)  sum = sum + a(i,j+2,k+1) + b(i,j+2,l) + c(i,k+1,l)  sum = sum + a(i,j+3,k+1) + b(i,j+3,l) + c(i,k+1,l)  sum = sum + a(i,j,k+2) + b(i,j,l) + c(i,k+2,l)  sum = sum + a(i,j+1,k+2) + b(i,j+1,l) + c(i,k+2,l)  sum = sum + a(i,j+2,k+2) + b(i,j+2,l) + c(i,k+2,l)  sum = sum + a(i,j+3,k+2) + b(i,j+3,l) + c(i,k+2,l)  sum = sum + a(i,j,k+3) + b(i,j,l) + c(i,k+3,l)  sum = sum + a(i,j+1,k+3) + b(i,j+1,l) + c(i,k+3,l)  sum = sum + a(i,j+2,k+3) + b(i,j+2,l) + c(i,k+3,l)  sum = sum + a(i,j+3,k+3) + b(i,j+3,l) + c(i,k+3,l)  sum = sum + a(i,j,k) + b(i,j,l+1) + c(i,k,l+l)  sum = sum + a(i,j+1,k) + b(i,j+1,l+1) + c(i,k,l+l)  sum = sum + a(i,j+2,k) + b(i,j+2,l+1) + c(i,k,l+l)  sum = sum + a(i,j+3,k) + b(i,j+3,l+1) + c(i,k,l+l)  sum = sum + a(i,j,k+1) + b(i,j,l+l) + c(i,k+1,l+1)  sum = sum + a(i,j+1,k+1) + b(i,j+1,l+1) + c(i,k+1,l+1)  sum = sum + a(i,j+2,k+1) + b(i,j+2,l+1) + c(i,k+1,l+1)  sum = sum + a(i,j+3,k+1) + b(i,j+3,l+1) + c(i,k+1,l+1)  sum = sum + a(i,j,k+2) + b(i,j,l+1) + c(i,k+2,l+1)  sum = sum + a(i,j+1,k+2) + b(i,j+1,l+1) + c(i,k+2,l+1)  sum = sum + a(i,j+2,k+2) + b(i,j+2,l+1) + c(i,k+2,l+1)  sum = sum + a(i,j+3,k+2) + b(i,j+3,l+1) + c(i,k+2,l+1)  sum = sum + a(i,j,k+3) + b(i,j,l+1) + c(i,k+3,l+1)  sum = sum + a(i,j+1,k+3) + b(i,j+1,l+1) + c(i,k+3,l+1)  sum = sum + a(i,j+2,k+3) + b(i,j+2,l+1) + c(i,k+3,l+1)  sum = sum + a(i,j+3,k+3) + b(i,j+3,l+1) + c(i,k+3,l+1)  sum = sum + a(i,j,k) + b(i,j,l+2) + c(i,k,l+2)  sum = sum + a(i,j+1,k) + b(i,j+1,l+2) + c(i,k,l+2)  sum = sum + a(i,j+2,k) 4 b(1,j+2,l+2) + c(i,k,l+2)  sum = sum + a(i,j+3,k) + b(i,j+3,l+2) + c(i,k,l+2)  sum = sum + a(i,j,k+1) + b(i,j,l+2) + c(i,k+1,l+2)  sum = sum + a(i,j+1,k+1) + b(i,j+1,l+2) + c(i,k+1,l+2)  sum = sum + a(i,j+2,k+1) + b(i,j+2,l+2) + c(i,k+1,l+2)  sum = sum + a(i,j+3,k+1) + b(i,j+3,l+2) + c(i,k+1,l+2)  sum = sum + a(i,j,k+2) + b(i,j,l+2) + c(i,k+2,l+2)  sum = sum + a(i,j+1,k+2) + b(i,j+1,l+2) + c(i,k+2,l+2)  sum = sum + a(i,j+2,k+2) + b(i,j+2,l+2) + c(i,k+2,l+2)  sum = sum + a(i,j+3,k+2) + b(i,j+3,l+2) + c(i,k+2,l+2)  sum = sum + a(i,j,k+3) + b(i,j,l+2) + c(i,k+3,l+2)  sum = sum + a(i,j+1,k+3) + b(i,j+1,l+2) + c(i,k+3,l+2)  sum = sum + a(i,j+2,k+3) + b(i,j+2,l+2) + c(i,k+3,l+2)  sum = sum + a(i,j+3,k+3) + b(i,j+3,l+2) + c(i,k+3,l+2)  sum = sum + a(i,j,k) + b(i,j,l+3) + c(i,k,l+3)  sum = sum + a(i,j+1,k) + b(i,j+1,l+3) + c(i,k,l+3)  sum = sum + a(i,j+2,k) + b(i,j+2,l+3) + c(i,k,l+3)  sum = sum + a(i,j+3,k) + b(i,j+3,l+3) + c(i,k,l+3)  sum = sum + a(i,j,k+1) + b(i,j,l+3) + c(i,k+1,l+3)  sum = sum + a(i,j+1,k+1) + b(i,j+1,l+3) + c(i,k+1,l+3)  sum = sum + a(i,j+2,k+1) + b(i,j+2,l+3) + c(i,k+1,l+3)  sum = sum + a(i,j+3,k+1) + b(i,j+3,l+3) + c(i,k+1,l+3)  sum = sum + a(i,j,k+2) + b(i,j,l+3) + c(i,k+2,l+3)  sum = sum + a(i,j+1,k+2) + b(i,j+1,l+3) + c(i,k+2,l+3)  sum = sum + a(i,j+2,k+2) + b(i,j+2,l+3) + c(i,k+2,l+3)  sum = sum + a(i,j+3,k+2) + b(i,j+3,l+3) + c(i,k+2,l+3)  sum = sum + a(i,j,k+3) + b(i,j,l+3) + c(i,k+3,l+3)  sum = sum + a(i,j+1,k+3) + b(i,j+1,l+3) + c(i,k+3,l+3)  sum = sum + a(i,j+2,k+3) + b(i,j+2,l+3) + c(i,k+3,l+3)  sum = sum + a(i,j+3,k+3) + b(i,j+3,l+3) + c(i,k+3,l+3)  end do end do do j = j, n  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  sum = sum + a(i,j,k+1) + b(i,j,l) + c(i,k+1,l)  sum = sum + a(i,j,k+2) + b(i,j,l) + c(i,k+2,l)  sum = sum + a(i,j,k+3) + b(i,j,l) + c(i,k+3,l)  end do end do end do do k = k, n do j = 1, n-3, 4  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  sum = sum + a(i,j+1,k) + b(i,j+1,l) + c(i,k,l)  sum = sum + a(i,j+2,k) + b(i,j+2,l) + c(i,k,l)  sum = sum + a(i,j+3,k) + b(i,j+3,l) + c(i,k,l) end do end do do j = j, n do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l) end do end doend dodo k = 1, n-3, 4 do j = j, n do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l+1) + c(i,k,l+1)  sum = sum + a(i,j,k+1) + b(i,j,l+1) + c(i,k+1,l+1)  sum = sum + a(i,j,k+2) + b(i,j,l+1) + c(i,k+2,l+1)  sum = sum + a(i,j,k+3) + b(i,j,l+1) + c(i,k+3,l+1)  end do end do end do do k = k,n do j = 1, n-3, 4  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l+1) + c(i,k,l+1)  sum = sum + a(i,j+1,k) + b(i,j+1,l+1) + c(i,k,l+1)  sum = sum + a(i,j+2,k) + b(i,j+2,l+1) + c(i,k,l+1)  sum = sum + a(i,j+3,k) + b(i,j+3,l+1) + c(i,k,l+1)  end do end do do j = j, n  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l+1) + c(i,k,l+1)  end do end do end do do k = 1, n-3, 4 do j = j, n  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l+2) + c(i,k,l+2)  sum = sum + a(i,j,k+1) + b(i,j,l+2) + c(i,k+1,l+2)  sum = sum + a(i,j,k+2) + b(i,j,l+2) + c(i,k+2,l+2)  sum = sum + a(i,j,k+3) + b(i,j,l+2) + c(i,k+3,l+2)  end do end do end dodo k = k, n do j = 1, n-3, 4 do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l+2) + c(i,k,l+2)  sum = sum + a(i,j+1,k) + b(i,j+1,l+2) + c(i,k,l+2)  sum = sum + a(i,j+2,k) + b(i,j+2,l+2) + c(i,k,l+2)  sum = sum + a(i,j+3,k) + b(i,j+3,l+2) + c(i,k,l+2)  end do end do do j = j, n  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l+2) + c(i,k,l+2)  end do end do end do do k = 1, n-3, 4 do j = j, n  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l+3) + c(i,k,l+3)  sum = sum + a(i,j,k+1) + b(i,j,l+3) + c(i,k+1,l+3)  sum = sum + a(i,j,k+2) + b(i,j,l+3) + c(i,k+2,l+3)  sum = sum + a(i,j,k+3) + b(i,j,l+3) + c(i,k+3,l+3)  end do end do end do do k = k, n do j = 1, n-3, 4  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l+3) + c(i,k,l+3)  sum = sum + a(i,j+1,k) + b(i,j+1,l+3) + c(i,k,l+3)  sum = sum + a(i,j+2,k) + b(i,j+2,l+3) + c(i,k,l+3)  sum = sum + a(i,j+3,k) + b(i,j+3,l+3) + c(i,k,l+3)  end do end do do j = j, n  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l+3) + c(i,k,l+3)  end do end do end doend dodo l = l, n do k = 1, n-3, 4 do j = 1, n-3, 4  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  sum = sum + a(i,j+1,k) + b(i,j+1,l) + c(i,k,l)  sum = sum + a(i,j+2,k) + b(i,j+2,l) + c(i,k,l)  sum = sum + a(i,j+3,k) + b(i,j+3,l) + c(i,k,l)  sum = sum + a(i,j,k+1) + b(i,j,l) + c(i,k+1,l)  sum = sum + a(i,j+1,k+1) + b(i,j+1,l) + c(i,k+1,l)  sum = sum + a(i,j+2,k+1) + b(i,j+2,l) + c(i,k+1,l)  sum = sum + a(i,j+3,k+1) + b(i,j+3,l) + c(i,k+1,l)  sum = sum + a(i,j,k+2) + b(i,j,l) + c(i,k+2,l)  sum = sum + a(i,j+1,k+2) + b(i,j+1,l) + c(i,k+2,l)  sum = sum + a(i,j+2,k+2) + b(i,j+2,l) + c(i,k+2,l)  sum = sum + a(i,j+3,k+2) + b(i,j+3,l) + c(i,k+2,l)  sum = sum + a(i,j,k+3) + b(i,j,l) + c(i,k+3,l)  sum = sum + a(i,j+1,k+3) + b(i,j+1,l) + c(i,k+3,l)  sum = sum + a(i,j+2,k+3) + b(i,j+2,l) + c(i,k+3,l)  sum = sum + a(i,j+3,k+3) + b(i,j+3,l) + c(i,k+3,l)  end do end do do j = j, n  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  sum = sum + a(i,j,k+1) + b(i,j,l) + c(i,k+1,l)  sum = sum + a(i,j,k+2) + b(i,j,l) + c(i,k+2,l)  sum = sum + a(i,j,k+3) + b(i,j,l) + c(i,k+3,l)  end do end do end do do k = k, n do j = 1, n-3, 4  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  sum = sum + a(i,j+1,k) + b(i,j+1,l) + c(i,k,l)  sum = sum + a(i,j+2,k) + b(i,j+2,l) + c(i,k,l)  sum = sum + a(i,j+3,k) + b(i,j+3,l) + c(i,k,l)  end do end do do j = j, n  do i = 1, n  sum = sum + a(i,j,k) + b(i,j,l) + c(i,k,l)  end do end do end doend do______________________________________ 
    
     Referring next to Table D, psuedo-code illustrating operations preferred in the carrying out of the present invention are shown. This psuedo-code is also illustrated in FIG. 3. 
     
                       TABLE D______________________________________INPUTS:1.  LOOP[l]. . . LOOP[k], a perfect nest of k loops, numbered from    outermost to innermost, in an optimizer intermediate representation.    The notation, &#34;do i.sub.j = lb.sub.j, ub.sub.j, inc.sub.j &#34;, denotes    the contents    of statement LOOP[j] with index variable i.sub.j, lower bound    lb.sub.j, upper bound ub.sub.j, and increment inc.sub.j.2.  UF[j] &gt;= 1, unroll factor for each LOOP[j].3.  COUNT[j], constant value or symbolic expression for number of    iterations executed by LOOP[j], where UF[j] is assumed to be less    than or equal to COUNT[j] if COUNT[j] is a constant. It is well    known by those skilled in the art to derive COUNT[j] from ib.sub.j    (lower bound of j), ub.sub.j (upper bound of j), and inc.sub.j    (increment of j).OUTPUT:1.  Updated intermediate representation of the unrolled loops to reflect    the loop unrolling transformation specified by UF[l], . . ., UF[k].METHOD:Set next.sub.-- parent = parent of LOOP[j] in intermediaterepresentation, the  original loop nest;Detach subtree rooted at LOOP[l] from next.sub.-- parent/* this subtree  is used as the source for generating copies of the original loop body  in the unrolled and remainder loops */;Set unrolled.sub.-- body = copy of body of innermost loop, LOOP[k];for(j = l; j &lt;= k; j++) { Set current.sub.--parent = next.sub.-- parent; /* STEP 1: Expand unrolled.sub.-- body by factor UF[j] for index i.sub.j*/ Set new.sub.-- unrolled.sub.-- loop.sub.-- body = copy ofunrolled.sub.-- body; for (u = l; u &lt;= UF[j] - 1; u++) {  Set one.sub.-- copy = copy of unrolled.sub.-- body;  replace all references of loop index variable &#34;i.sub.j &#34; in one.sub.--copy by  &#34;i.sub.j + inc.sub.j *u&#34;;  append one.sub.-- copy to end of new.sub.-- unrolled.sub.-- loop.sub.--body; } Delete unrolled.sub.-- body and set unrolled.sub.-- body = new.sub.-- unrolled.sub.-- loop.sub.-- body; /* STEP 2: Adjust header for unrolled loop j */ Construct the remainder expression er.sub.j, = mod(COUNT[j], UF[j]); if( COUNT[j] is constant and COUNT[j] = UF[j]) {  /* loop j is to be completely unrolled */  Construct the statement, &#34;i&#34; = lb.sub.j &#34;, call it next.sub.-- parent,  and make it the first  (leftmost) child of current.sub.-- parent; } else {  Make a copy of the LOOP[j] statement, call it next.sub.-- parent,change  it to &#34;do i.sub.j = lb.sub.j, ub.sub.j - er.sub.j *inc.sub.j,  UF[j]*inc.sub.j &#34;, and make it a child of  current.sub.-- parent; } /* STEP 3: Generate remainder loop sub-nest, if necessary */ if (er.sub.j is not a constant 0) {  Set tree.sub.-- copy = copy of subtree rooted at LOOP[j];  change the outermost statement in tree.sub.-- copy to  &#34;do i.sub.j = ub.sub.j - (er.sub.j -l)*inc.sub.j, ub.sub.j, inc.sub.j&#34;;  Make tree.sub.-- copy a child of current.sub.-- parent; } } /* for (j = . . . ) */Make unrolled.sub.-- body a child of next.sub.-- parent;Delete subtree rooted at LOOP[l] (original loop nest);______________________________________ 
    
     Referring next to FIG. 3 through FIG. 6, flowcharts illustrating operations preferred in carrying out the present invention are shown. In the flowcharts, the graphical conventions of a diamond for a test or decision and a rectangle for a process or function are used. These conventions are well understood by those skilled in the art, and the flowcharts are sufficient to enable one of ordinary skill to write code in any suitable computer programming language. These flowcharts correspond to the pseudo-code of Table D. 
     Referring now to FIG. 4, the process of the invention, generally referred to as 400, begins at process block 405. Thereafter, process block 410 sets a next --  parent equal to the parent of LOOP[1] in an intermediate representation, and process block 415 detaches a subtree rooted at LOOP[1] from the next --  parent. FIG. 5 illustrates the tree structure 500 for the original intermediate representation of the loops to be unrolled comprising a parent 510 of LOOP[1] 520, and k perfectly nested loops comprising LOOP[1] 520, LOOP[2] 530, and LOOP[3] 540. Next, process block 420 sets an unrolled --  body 550 equal to a copy of a body of an innermost loop, LOOP[k]. Then, process block 425 begins a for loop with j initialized to 1 and incremented by j++ while j is less than or equal to k. Within this loop, process block 430 sets a current --  parent equal to the next --  parent, and then performs three steps. Step one, performed by process block 435, expands the unrolled --  body by a factor UF[j] for index i j . FIG. 6 illustrates an unrolled body 600 produced by process block 435 after j=1, and FIG. 7 illustrates an unrolled body 700 produced by process block 435 after j=2. Step two, performed by process block 440, adjusts a header for an unrolled loop j. Step three, performed by process block 445, generates a remainder loop sub-nest, if necessary. Thereafter, process block 450 increments the for loop index j by j++, and then decision block 455 determines if the for loop is completed (if j less than or equal to k). If the for loop is not completed, then processing loops back to process block 430 to process the next --  parent. 
     Returning now to decision block 455, if the for loop is completed (if j not less than or equal to k), then process block 460 makes the unrolled --  body a child of next --  parent, and process block 465 deletes the subtree rooted at LOOP[1]. The process then ends at process block 470. 
     Referring now to FIG. 8, an expansion of process block 435 of FIG. 4, generally referred to as 800, is illustrated. This process 800 performs the first step of expanding the unrolled --  body by a factor UF[J] for index i j . The process 800 begins at process block 805, and thereafter, process block 810 sets a new --  unrolled --  loop --  body equal to a copy of the unrolled --  body. Then, process block 815 begins a for loop with u initialized to 1 and incremented by u++ while u is less than or equal to UF[j]-1. Within this loop, process block 820 sets one --  copy equal to the copy of unrolled --  body; process block 825 replaces all references of loop index variable &#34;i j  &#34; in one --  copy by &#34;i j  +inc j  *u&#34;; and process block 830 appends one --  copy to the end of new --  unrolled --  loop --  body. Process block 835 then increments the for loop index u by u++, and decision block 840 determines if the for loop is completed (if u less than or equal to UF[j]-1). If the for loop is not completed, then processing loops back to process block 820 to again process one --  copy. 
     Returning now to decision block 840, if the for loop is completed (if u not less than or equal to UF[j]-1), then process block 845 deletes the unrolled --  body and sets unrolled --  body equal to the new --  unrolled --  loop --  body, and then processing returns through process block 850 to continue processing process block 440 of FIG. 4. 
     Referring now to FIG. 9, an expansion of process block 440 of FIG. 4, generally referred to as 900, is illustrated. This process 900 performs the second step of adjusting the header for an unrolled loop j. The process 900 begins at process block 905, and thereafter, process block 920 constructs the remainder expression erj which is equal to mod(COUNT[j], UF[j]). Decision block 930 then determines if COUNT[j] is a constant equal to UF[j]. If COUNT[j] is a constant equal to UF[j], then loop j is to be completely unrolled and process block 940 constructs the statement, &#34;i j  =lb j  &#34;, calls it next --  parent, and makes it a child of current --  parent, and then processing returns through process block 950 to continue processing process block 445 of FIG. 4. 
     Returning now to decision block 930, if COUNT[j] is not a constant equal to UF[j], then process block 960 makes a copy of the LOOP[j] statement; calls it next --  parent; changes it to &#34;do i j  lb j , ub j  -er j  *inc j , UF[j]*inc j  &#34;; and makes it a child of current --  parent. Processing then returns through process block 950 to continue processing process block 445 of FIG. 4. 
     Referring now to FIG. 10, an expansion of process block 445 of FIG. 4, generally referred to as 1000, is illustrated. This process 1000 performs the third step of generating a remainder loop sub-nest, if necessary. The process 1000 begins at process block 1010, and thereafter, decision block 1020 determines if er j  is not a constant equal to 0. If er j  is not a constant equal to 0, then process block 1030 sets tree --  copy equal to the copy of subtree rooted at LOOP[j]; process block 1040 changes the outermost statement in tree --  copy to &#34;do i j  =ub j  -(er j  -1)*inc j , ub j , inc j  &#34;; and process block 1050 makes tree --  copy a child of current --  parent. Processing then returns through process block 1060 to continue processing process block 450 of FIG. 4. 
     Returning now to decision block 1020, if er j  is a constant equal to 0, then processing returns through process block 1060 to continue processing process block 450 of FIG. 4. 
     Referring now to FIG. 11, a block diagram illustrates a computer system 1100 used in performing the method of the present invention, forming part of the apparatus of the present invention, and which may use the article of manufacture comprising a computer-readable storage medium having a computer program embodied in said medium which may cause the computer system to practice the present invention. The computer system 1100 includes a processor 1102, which includes a central processing unit (CPU) 1104, and a memory 1106. Additional memory, in the form of a hard disk file storage 1108 and a computer-readable storage device 1110, is connected to the processor 1102. Computer-readable storage device 1110 receives a computer-readable storage medium 1112 having a computer program embodied in said medium which may cause the computer system to implement the present invention in the computer system 1100. The computer system 1100 includes user interface hardware, including a mouse 1114 and a keyboard 1116 for allowing user input to the processor 1102 and a display 1118 for presenting visual data to the user. The computer system may also include a printer 1120. 
     Although the present invention has been particularly shown and described with reference to a preferred embodiment, it will be understood by those skilled in the art that various changes in form and detail may be made without departing from the spirit and the scope of the invention.