The person charging this material is re- sponsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN OEC 1 4 1976 DEC12^CG IP i o : )B2 AUG 2 9 1997 L161 — 0-1096 UIUCDCS-R-72-57^ jTlUJi ITERATIVE AND DIRECT METHODS FOR SOLVING POISSON'S EQUATION AND THEIR ADAPTABILITY TO ILLIAC IV by James H. Ericksen December 20, 1972 ^ DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN URBANA, ILLINOIS THE LIBRARY OF THF UIUCDCS-R-72-57^ ITERATIVE AND DIRECT METHODS FOR SOLVING POISSON'S EQUATION AND THEIR ADAPTABILITY TO ILLIAC IV by James H. Ericksen Department of Computer Science University of Illinois at Urb ana-Champaign Urbana, Illinois 6l801 December 20, 1972 This work was supported in part by the Advanced. Research Project Agency of the Department of Defense and was monitored by the U. S. Army Research ' Office-Durham, under Contract No. DAH CO^-72-C-OOOl, and supported in part by the Atmospheric Sciences Section, National Science Foundation, NSF Grant J GA-31^07. Portions of this work were submitted in partial fulfillment of the requirements for the degree of Master of Science in Computer Science in the Graduate College of the University of Illinois at Urb ana-Champaign, June, 1972. Digitized by the Internet Archive in 2013 http://archive.org/details/iterativedirectm574eric ACKNOWLEDGEMENT The author wishes to express his gratitude to Professor Michael S. Sher and Mrs. Masako Ogura for suggestions and comments throughout this project. The author is grateful to Dr. Walter L. Heimerdinger who supplied information dealing with timing on ILLIAC IV. The author wishes to thank Mrs. Nancy Freece for typing this manuscript. Finally, the author is indebted to the Center for Advanced Computation for providing the opportunity to carry out this study. ABSTRACT This paper examines iterative and direct methods for solving Poisson's iquation with regard to their adaptation to ILLIAC IV. SOR, SLOP, ADI, and FACR ,re programmed in GLYPNIR. Detailed suggestions on ASK code for these methods re also supplied. FACR, Fourier Analysis and Cyclic Reduction, is the fastest method on ■ectangular meshes. SOR, Successive Over Relaxation, seems to be the most iromising for nonrectangular meshes. The methods are between thirty and forty - ive times faster on ILLIAC IV than on a serial machine with speed equal to one )f the ILLIAC IV PEs (Processing Elements). TABLE OF CONTENTS Page 1. POISSON'S EQUATION IN MATRIX FORM 1 2. IMPLEMENTATION OF ITERATIVE METHODS ON ILLIAC IV 6 2.1 Introduction to SLOR and SOR 6 2.2 Introduction to ADI 9 2.3 Implementation of SOR on ILLIAC IV 11 2.3.1 A Parallel Processor's Effect on the Algorithm. ... 11 2.3.2 Parallelisms in SOR 12 2.3.3 SOR Using Straight Storage l6 2.3. 1 ! SOR Using Odd-Even Storage 20 2.3-5 SOR Machine Efficiency 23 2.1+ Implementation of SLOR on ILLIAC IV 2k 2.U.1 Parallelism in SLOR 2k 2.U.2 Storage for SLOR 25 2.U.3 Improving a Line by the Cuthill-Varga Method 25 2.k.k SLOR Implementation 28 2.5 Implementation of ADI on ILLIAC IV 31 2.5.1 Thomas' Method on ILLIAC IV 32 2.5.2 Skewed Storage 33 2.6 Summary of the Results for Iterative Methods 39 3. IMPLEMENTATION OF DIRECT METHODS ON ILLIAC IV kk 3.1 Introduction to Hockney's Direct Method (FACR) kk 3.2 Odd/Even Reduction and Odd Column Solution kG 3.3 CRED k9 3.k Fourier Analysis and Synthesis 53 3-5 Implementation of MFACR 60 page 3.6 Implementation of FACR 62 3.7 Summary of the Results on Direct Methods 62 k. USE OF ILLIAC DISK FOR NON-CORE CONTAINED MESHES 65 k.l ILLIAC IV I/O System 65 k.2 I/O for the Bernard-Rayleigh Convection Problem , 65 5. CONCLUSIONS 71 REFERENCES 7 3 APPENDIX A. TIMING METHODOLOGY A-l APPENDIX B. MAJOR STEPS OF FACR AND MFACR FOR DIRICHLET'S BOUNDARY CONDITIONS B-l APPENDIX C. SUCCESSIVE (POINT) OVER-RELAXATION METHOD IN MSOR. . C-l APPENDIX D. SUCCESSIVE LINE OVER-RELAXATION METHOD IN GLYPNIR, SLOR D-l APPENDIX E. PEACEMAN-RACHFORD ADI IN GLYPNIR E-l APPENDIX F. HOCKNEY'S DIRECT METHOD, MODIFIED, MFACR F-l LIST OF TABLES Page Table 1. Parallelism of SOR 13 Table 2. Timing of an Execution of Phase 1 SOR using Straight Storage 19 Table 3. Timing of an Execution of Phase 1 SOR using Odd-Even Storage 22 Table k. Timing of an Execution of Phase 1 SOR 30 Table 5- Timing of an Execution of Phase 1 ADI 38 Table 6. A Numerical Study of the Methods under Discussion 39 Table 7- Summary of Time and Storage Requirements for the Iterative Methods on a m by n Mesh 1+0 Table 8. Comparison of Three Methods to Solve Tridiagonal Matrix Equa- tions for Dirichlet's or Neumann's boundary conditions ... 52 Table 9. Preparation of the Data for the Fast Fourier Transform ... 56 Table 10. The Complex Fast Fourier Transform 57 Table 11. Final Step in Cooley's Fourier Analysis and Synthesis. ... 58 Table 12. CRED 59 Table 13. A Comparison of Methods with Respect to Time 72 Table lU. Assignment Statement Notation A-2 Table 15 . Assumptions in Timing A-2 LIST OF FIGURES Page Figure 1. The Mesh ■ 2 Figure 2. Straight Storage of Row j of U 17 Figure 3. Odd-Even Storage Data Allocation of Row j of U Where m is Even 21 Figure h. A 96 "by 96 Mesh in Skewed Storage 34 Figure 5. Storage Schemes Used on Rows k and 5 of a 96 by 96 Mesh if Odd/Even Reduction is Used 48 Figure 6. A Disk Map of 512 by 512 M es hes TO 1. POISSON'S EQUATION IN MATRIX FORM We will consider Poisson's equation with Dirichlet boundary condi- tions defined on a rectangular region. This equation can be written as t (X, T ) = ^M + ^%^ (1) dY 2 ox 2 where the value of U is given at the boundaries X = 0, Y = 0, X = (m-l)h x and Y = (n-l)h We denote U((i-l)h , (d-l)h ) by U t('(i-l)h , (j-l)h ) by \|/ . . and error of order p by 0(p). By dividing the region into mesh points such that the distance between mesh points in the vertical direction is h and in the horizontal direction is h we get Figure l.a. The region y x in Figure l.a can be rotated 90 clockwise to obtain the mesh U in Figure l.b where U. . is the mesh point in the i — row and the ,i — column. 1*3 By use of Taylor's Theorem we obtain: 2 3 k 1 h > 2 tt h ^3 TT n ^k„ 3.U/. • \ xa U/. . \ xd U/. . n xd U /. . \ 1+1, J 1,3 x 9X ^ ^ 3- ^ x O 4. ^-yt 2 3 !+ 1 tt tt t- oU/. ,\ xdU/. .x x d U /r. ,\ xdU/. .n / s y 0,n- 1 l,n- ■ 1 2,n- 1 m-3,n- ■ 1 m-2,n-l m-l,n-l 0,n- 2 l,n- 2 2,n- .2 T" m-3,n- ■2 m-2,n-2 m-l.n-2 < • • i l* • it 0,n- 3 l,n- 3 2,n- 3 m-3,n- ■3 m-2,n-3 m-l.n-3 i • • l ,♦ • i i 0,2' 1,2 2,2 m-3,2 m-2,2 m-1,2 i » ' • • r I* • <► 0,1 • 1,1 1 1 2,1 U m-3,1 * m-2,1 ■ jn-1,1 t 1 r h — < 0,0 1,0. 2,0 I m-3,0 m-2,0 ^ . »-i*o / J ' > 1* h A a) The rectangular region with the mesh points. U l,l' U l,2> U l,3' ' l,n-l' l,n U 2,V | U 2,2> U 2,3' ' ' ' U 2,n-1' { U 3,l' | U 3,2> U 3 , 3 ' * ' * U 3,n-1' U m-l,l'i U m-1,2 Vl,3" U U U 111,1' m,2' m,3' U m-l,n.-l' U u, 2,n U, 3,n U m-l,n U m,n-l m,n b) The mesh points with separation of . boundary and interior mesh points, Figure 1. The Mesh 8 2 U ,. . Thus I (i ^ } = "5 ^ U i i + l + U i i 1 " 2U i i) + °( h v ) ( 2 ) <= -u^ i,o+i 1,0-1 i,o y 5Y and £-§(!,;)) . i (U + U - OTJ ) + O(^) (3) 85T h x Substituting (2) and (3) into (l) we obtain % (U. . , + U. . , - 2U. .) + ■— (U. . . + U. n • - 2U. -) ss \|r. . • (if) h 2 v i,J+l 1,3-1 1,0 h 2 1+1,0 1-1,3 i,J 1,0 y x This is the equation for any interior mesh point U. .. The values of the 1,0 mesh points on the boundary are given, and require no equations. Let us take all the equations for the interior mesh points and place the constant terms on the right hand side of the equations. For example, the equation for tL . where 3 < o < n-2 will be 2, J J _ ^ _ y xx The equation for U p „ will be P (U 2,3 - 2U 2,2' + k CU 3,2 " Z \2 ] ~ *2,2 " k V " 72 U 2,l y X . X y Combining the constant terms into one term, M. ., for each U. . we obtain i>J 1,3 ;. M i,j " *i,J " h (U i,o + l W + U l,o-l W 5 " h 1*1+1,3 W y x « ! + Vi/i.i,i) ^ i where fi _ { if k + i where b^ £ - ( x lf k „ ^ • Consider the internal mesh points as a vector U_ in which- the order of the mesh points is as follows: 1) The first interior point in the first row becomes the first element of the vector. 2) All interior points of row i are placed in the vector in the same order they appear in row i. 3) The (n-l) — element of the i — row is followed by the first interior point of the following row, where 2 < i < m_2. The transpose of U-j- equals (U 2,2> U 2,3' '."' U 2,n-i; U 3,2' U 3,3' '"' \a-V ' ' m-1,2 m-1,3' U m-l,n-l) Using this order we assemble the interior mesh point equations and consider them as the matrix equation AU = M -where M is the vector of constant terms and A is a matrix made up of the coefficients of the mesh points. The matrix A is a (m-2)(n_2) by (m-2)(n-2) block tridiagonal matrix of the following form: A = B C C B C C B C B G 1 (6) where C = al h-2' +1 2 h X and B is a (n-2) by (n-2) tridiagonal matrix of the form "^ d e d \ s \ \ \ \ \ \ \ \ v x \ \ d e d (6.1) J ■where d = — and e = - (2a + 2d), h y There are two general approaches used in solving this system of linear equations, direct methods and iterative methods.* In this study we have limited our discussion to three popular iterative methods and one direct method: the successive over-relaxation method (SOR), the successive line over-relaxation method (SLOR), the alternating direction implicit method (ADI), and Hockney's direct method (FACR). In the discussions of FACR three types of boundary conditions will be examined: periodic, Neumann and Dirichlet. j The discussion of iterative methods considers only Dirichlet boundary conditions , No matter which method is used in solving the equations the lower bound on the error is (max(h 2 , h 2 )). This is due to the error terms in (2) and (3) y x' 2. IMPLEMENTATION OF ITERATIVE METHODS ON ILLIAC IV 2.1 Introduction to SLOR and SOR Iterative methods have one thing in common. They each begin with an initial guess to the solution and improve upon the guess with each iteration. Thus the following notation is useful: The value of UL after the &21 iteration is denoted by u| ^ . U^ j is the initial guess and if the method converges then Lim (U - U ) = 0. Matrix A can be divided into three matrices such that A = D - E - F where D is block diagonal, E is strictly lower triangular and F is strictly upper triangular. Thus AU = M can be written as DILj. - (E + P) llj. = M. From this we get the iterative method: U^ +l) = D" 1 (E + F) TjW + if 1 M (7) The matrix equation can also be grouped to obtain U (i+l) = (D-E) -1 FU^ } + (D-E)" 1 M (8) The asymptotic convergence rate of (8) is twice as fast as (7) [Todd, Page 391 J • This is due to the fact that the matrix has Young's property A [Wachspress, Page 102]. (8) is a successive method while (7) is a ** simultaneous method. By defining J = — (D - u>E) and H = — (wF + (l-w)D) for u^Ove get to to u JU-j. = u)HU + oM from AU = M = (J-H)U . * Improvement is measured by a reduction in the error vector. ** Successive methods use new information as soon as it is available while simultaneous methods use old values for the entire iteration. We can write wJU = wHU + wM as (D-wE)U. (ihl) = (goF + (l-w)D) \j[ £) + ojM (9) Note that (8) and (9) are equivalent if id = 1. When < w < 2 and A is positive definite, (9) converges. If 1< oo s.2, then (9) is referred to as an over-relaxation method [Forsythe and Wasow, Page 26l ] . With different values of io, the convergence rate of (9) is altered. For a 2 single relaxation parameter, it can be shown that to = ; — , b 1 +/l-p(3) where 3 = D (E+F), the Jacobi matrix, and p(8) is the spectral radius of 3, gives the fastest rate of convergence [Young, Page 169 ] . There are two common ways in which equations (7 - 9) are implemented • The first gives point iterative methods by letting D equal the diagonal of A. Thus (7) becomes Jacobi 's method, (8) becomes the formula for succes- sive point relaxation iterative method (SR) and (9) becomes the successive point over relaxation iterative method (SOR). If we let D = where B is defined in (6.1) then we will get line iterative methods. (7) becomes the simultaneous line iterative method, (8) becomes the successive line relaxation iterative method (SLR) and (9) becomes the successive line over-relaxation method (SLOR) [Varga, Page 199] . To give the reader a better understanding of the way these methods work, we write the individual equations for the interior mesh points U. ., where 3 < i < m-2 and 3 < < n -2 for each method. First we de- 1, fine h, v, and g as follows: ^2 .2 u 2 v 2 h h h h h = o Y o , v = p 2 o > and g = -— Z_ ' 2(h 2 + h 2 ) 2(h 2 + h 2 ) 2(h 2 + h 2 ) v x y x x y x y y Jacobi 's method (simultaneous point iterative method) „U+1) /..(i) TT (i) v . , TT (i) TT (i) V U: . = v (U: . , + U. . , ) + h (U. t . + U. t .) - g M. 1,0 1,0+1 1,0-1 v 1+1,0 1-1,0 1,. Simultaneous line iterative method TT (i+l) , TT U+l) tt(^+1)n i. /tt(^) ttU) N U. . = v (U: . i + U. . ' )+ h (U: : . + U. ' .) - g M. 1,0 1,0+1 1,0-1 1+1,0 i-l,0 i,. Successive point relaxation method (SR) (10) Note we have excluded the first row, last row, first column, and the last column of the interior mesh points to avoid boundary conditions. These points will be discussed in another section. Successive line relaxation method (SLR) (Ml), . T (u (i + D + u^D, + h (U U) + ^1), . M . . . (li: i*3 1*3+1 1*3-1 1+1*0 1-1*3 1*3 Successive point over-relaxation (SOR) u (i+1) . ,» v (u^. , + a** 1 ]) + » h (u^ . + u"; 1 ?) .,,«. . 1,3 1,3+1 1,3-1 1+1,3 1-1,3 1,3 + ^ "if]' • (12) Successive line over-relaxation (SLOR) TT (i+l) / TT U+l) TT (i+lK t, / TT (^) TT (<0+1)\ U. . = v (U. . ' + U. . ' + w h U. ' . + U: . . ) - w g M. . i*3 i*3+l 1*3-1 1+1*3 1-1*3 i*3 + (1-00) (u^) _ v (u^) + u^) )). (13) 1*3 1*3+1 1*3-1 2 . 2 Introdu ction to API Let us look at the equation we are solving once again, ivi^ji + i^M± _ T ( fl (1) as before we use Taylor' s Theorem and we get y and sfu 2 &X h x (1 '^72 (U i-l,j + U i + l,j - 2U i,^ + ^ (3) 10 Thus when we form the matrix equation AIL. = M where A is the matrix (6), we are considering a horizontal part of the equation and a vertical part of the equation. We could divide A into two matrices H and V such that H + V = A where V is the matrix containing the coefficients of the vertical equations (2). Both H and V can be placed in tridiagonal form with the ap- propriate permutation matrix. We can take the equation AU = HU + VU = M and obtain (H - ^jjDUj = M - (V + w^l)^ and (V - o^DUj = M - (H + 0)^1)1^ • Thus we can get the Peaceman-Rachford AD I scheme [Varga, Page 212]. uW^h.^-I^.^^^ (lU.l) U x (i+1) = (V - u^l)- 1 [M - (H f a) iv l)U^ +1 / 2) ] (14.2) The parameters, u^ and u^ y are defined in [Wachspress, Page 192]. It will help to have the corresponding equations for the individual mesh points, IL ., where 3 < i < n-2 and 3 < j < m-2. (^1/2) _J _1 (i ) (J ) 2_ (!) m h 2 y y x _ _l (u (ifl/2) uU+1/2),, ; h 2 lu i-l,j + u i + i,j ".• (15.D x ADI requires that A be positive definite. This can be obtained by using - AU j = " M or by using -u) £V and -co instead of to y and co . We use the latter technique. 11 u (i+l) = -1 [M -J: (u (i+1 /2) + U (i+V2) ) _ , _ _2, (if 1/2) _2> LM i,J h 2 lU i + l,j + U i-l,J > " ^ W iV 9 j U i,j (^ y+ -f) X ' J h^ ^^ ^^ ^ hx y y Examining (15-1) and (15.2) we note a strong resemblance to simultaneous column relaxation followed by simultaneous row relaxation. 2 . 3 Implementation of SOR on I LLIAC IV 2.3.1 A Parallel Processor's Effect on the Algorithm A parallel machine allows operations to be performed simultaneously which are done in separate time intervals on a conventional machine. This is referred to as overlap. The amount of overlap is dependent on the way in which the algorithm is programmed. The SOR algorithm supplies an il- lustration of how the flow of an algorithm is changed when programmed to maximize overlap on a parallel machine. Let us examine the equation for SOR closer: ^^-^i^O^-^^^-- ^ (12) + (1K0) U{ £ \ for 3 < i < m-2 and 3 < j < n-2. Note the interior points which are adja- cent to one or more boundary points are not included. Also note that for the above mesh points M . = Y .. The equation for U would be 1 >3 1 >J 2,2 where M = W u - u 2,2 X 2,2 h 2 U l,2 ^2 U 2,l x h y Thus M 2,2 " ■ * 2)2 " * U 2,l " h V 2 • 12 This enables us to rewrite (l6) as u 17 - * v (%3 + $z ,] + u h <%k + u i; s ' - u s ¥ 2 ,2 ^ To calculate U using (17) takes two more additions than using (.16) 2,2 but M is needed to use (l6). On a serial machine, one would use equation (16) to improve U . The choice is not so straightforward on 2>2 a parallel machine if when U is calculated in one processing element 2>2 (PE), U. . is calculated in another PE where 3 < i 3 i,J i,J 13 indicates a boundary point. The interior mesh points are divided into seven sets (l,2,.3'»«> 7) where set i contains all points labeled by i. To improve any point in set 1, we must first improve all the points in the union of all sets j such that j < i. In a computer with k PEs, one iteration could be completed in 7 time steps where a time step is the amount of time to improve a mesh point in one PE by improving one set in each time step. But the computer could im- prove 28 points in 7 time steps. To improve the efficiency of the method, we look at what could be done in each time step assuming we had done everything possible up to that time step. Time Step 1 2 3 k 5 6 7 8 9 10 11 What r 1 2 1 3 1 k 1 5 1 6 1 7 1 Can Be I 2 2 2 3 2 k 2 2 5 6 2 7 2 Done At I 3 2 3 3 3 U 3 5 3 6 3 7 3 That Time 1* 2 U 3 U I 5 2 5 5 U 3 5 x 6 i 1 Table 1. Parallelism of SOR £ th i denotes the i — improvement of the elements in set i. In Table i W e see that steps 1 to 5 improve less than 8 elements but 8 elements are improved at each time step following step 5. Thus if we Ik have a computer with 8 PEs we can compute U in 5 + 2^ time steps. If we assume that our initial guess is 1 , 2 ,' 3 , ^ , 5 > 6 , 7 , instead of 1,2,3,^,5,6,7 then we can start at step 6 and do every iteration in 2 time steps. This can be generalized for an arbitrary mesh as follows: First we group all the interior mesh points into two sets, ODD = {U. . I i + j is odd}, EVEN = {U. . | i +j is even} i, i,0 Time step 1 improves all the points in EVEN. Time Step 2 improves all the points in ODD. We will call this method modified successive over- relaxation (MSOR). Remember all the points in ODD can be calculated using only points from EVEN and all the points in EVEN can be calculated using only points from ODD. Thus we can write a two part algorithm for MSOR [Young, Page 271]. U< i+1) .„., (v (4. 1 \ n + U<« ,) + h (S\ . + U^ .) - g f. .) | 1,0 Ei 1,0+1 1,0-1 1-1,0 1+1,0 1,0 r (i) i,0 for all the points in EVEN + (1 ^ei> u i,j (18.1; 1,0 Di v 1,0+1 1,0-1 1,0+1 1,0-1 1,0 r (i) i,0 + (l ^Di } U L (18.2; for all the points in ODD. 15 th In Young's discussion of MSOR, the optimal w , and j) , for the £ iteration are calculated. We will limit our study to MSOR where to . Intuitively one would think MSOR and SOR have very close rates of con- at vergence. In fact, their "asymptotic rates of convergence" are equivalent. If A is a convergent n x n complex matrix, for all £ sufficiently large, the £ £ average rate of convergence for £ iterations, R(A ), is Lim R(A ) = £ -» °° -lnp(A) s R (A) [Varga, Page 67]. Thus for a sufficiently large £, the asymptotic rate of convergence equals the average rate of convergence. We know if P is a permutation matrix that p(PAP ) = p(A) [ Birkhof f and Maclane, Page 2^9]. By multiplying (9) "by P we get pu U+l) = p £,_<£)-! tcoF + ( lMjJ )D] p _1 pu (i) (19) The asymptotic rate of convergence for (19) equals -Ln p(p(Da J jE)" 1 (wP +(i^ ) )D}p- 1 ) = -Lnp( (D^oE)" 1 {wFf (l-w)D)) (20) which equals the asymptotic rate of convergence for SOR. This not only shows that SOR and MSOR have the same asymptotic convergence rate, but that SOR applied to any permutation of U gives the same asymptotic rate The negative of the logarithm of the spectral radius of a convergent matrix A is the asymptotic rate of convergence, R (A), for the matrix A [Varga, Page 67 1. 16 of convergence. Thus we can choose the ordering of U which best fits the machine being used. 2.3.3 SOR using Straight Storage Now we shall examine the implementation of SOR on ILLIAC IV [Rudsinski]. We will compare two storage schemes and examine their effect on machine effi- ciency and their time per execution. The first storage scheme, Straight Storage, stores all the elements of row i before any of the elements of row j if i < j . The elements of each row appear in the order they are found in the mesh. The first element of each row is stored in PE and the last element of the row is stored in PE I where I equals (n-l) mod 6k, n is the number of columns. Figure 2 should help clarify this storage scheme. How can we implement SOR using Straight Storage? We observe that the PEs which contain the odd elements of mesh row I contain the even elements of mesh row I + 1. This enables us to improve the odd elements of Rows I and I +1 simultaneously by using a PE integer index which accesses an element * The timing method is explained in Appendix A. The terms used in timing are defined there also. IT Row U, Row U 3+1 Row U j+K-1 Row U J+K PEM V PEM 1 PEM I U. D,65 u 3, n -I U j+1,1 • u. ,2 u. ,66 • u. ,n+l- -I u. 3- .1,2 • • U. ,1+1 U. 3 ,65+1 • U. J >n u. J- •-1,1+1 1 PEM 63 # U J,64 u o,i28 ; i ! i U. , c\ a+1,64 • Figure 2. Straight Storage of Row j of U in row I (i+l) in the even (odd) PEs when I is even. In a like manner we can access all the even elements of two consecutive rows. If we have an odd number of rows we can improve the odd elements of the last row and the even elements of the first row simultaneously using a similar technique. Given a mesh with m rows and n columns, we use the index described above so that we can improve two rows simultaneously. Each pair of rows can K stands for the number of rows of PEM required to store n words and I = (n-1) mod 6U. 18 be divided into K groups where K = Ln/6UJ if n mod 6k = otherwise K = [n/6i+J + 1. Using Phase 1 we can improve 6U elements at a time (n-2)/6UJ times and then use Phase 2 to improve the remaining interior points. If we use Phase 1 to improve the first 6U odd (even) interior points of a pair of lines then we must take into account that only 63 of them are found in the same pair of rows of PEM because of the boundary points. This means we must use special indices to reach the 6Uth odd (even) interior point. This requires no extra computer time because the number of times register X is changed is not affected. SOR requires not only the point to be improved but its four neighbors and the appropriate value of Y. For example: To improve U. . we need U. ., U ., U. , _, U. . , , U and \|r These values will be accessed by a combination of ACAR and Register X indexing. To do the indexing, K will be combined with the CU integer Cl-equal to the first row of PEM containing mesh points to be improved during this execution - and three PE integers. PI=(1,K,0,K,0,...,0,K), PR=(K,0,K,0,...,K,0,) and PL=(K+1, 1,K,0,K,0, . . . ,K,0) if we are improving the odd points; PI=(K,1,0,K,0,K,...,K,0), PR=(0,K,0,K,...,0,K) and PL=(l,K+l,0,K,0,K, . . . ,0,K) if we are improving the even points; * [pj equals k where p is a real number, k is an integer, and k< p < k-fl. ** Phase 1 and Phase 2 are defined in Appendix D. 19 INITIAL CONDITIONS Register Contents Register Contents $ X PI $ D3 WI $ DO H $ CO CI $ Dl V $ CI CM $ D2 W $ C2 CP STEP ASSIGNMENT OPERATIONS AND OVERLAP TIME IK NO. STATEMENTS CLOCKS 1 $A: = U[$X] (0) PE fetch 7 2 $C3: = $D3; $A: = $A*$C3 CU fetch Real multiplication 10 3 $S: = $A - \|f[$X] (0) Real subtraction (PE fetch overlapped by- multiplication) 7 k $A: = U[$X] (2) (PE fetch overlapped by- addition) 5 $A: = $A + U[$X] (1) Real addition and PE fetch ik 6 $X: = PL (PE fetch overlapped by addition) 7 $C3: = $D0; $A: = $A*$C3 CU fetch Real multiplication 10 8 $R: = U[$X] (0) (PE fetch overlapped by multiplication) 9 $A: = $A+ $S Real addition 7 10 $X: = PR (PE fetch overlapped by addition) 11 $B: = RTL (1,,$R) Route 3 12 $R: = U[$X] (0) (PE fetch partially overlapped by Route) 1+ 13 $S: = $A Load from PE Register 1 Ik $A: = RTR (1,,$R) Route 3 15 $A: = $A + $B Real addition 7 16 $C3: = $Dlj $A: = $A*$C3 (CU fetch overlapped by addition) Real multiplication 9 17 $X: = PI (PE fetch overlapped by multiplication) 18 $A: = $A + $S R'^al addition 7 19 $C3: = $D2; $A: = $A*$C3 (CU fetch overlapped by addition) Real multiplication 9 20 U[$X] (0): = $A PE store 7 TOTAL TIME 105 Table 2. Timing of an Execution of Phase 1 SOR using Straight Storage 20 Let W = wg, V = v/g, H = h/g, and Wl = -~ where g, h, and v are defined in 1.2. Now (IT) can be written in GLYPNIR as follows: CP: = C + K; LOOP CI: = C, K + K, CK DO BEGIN CM: = CI - K; CP: = CI + K; (21) U [PI + CI]: = W*(V*(RTR(1,,U[PR+CT])+ RTL(l,,U[PL+Cl] ) ) + H*(U[PI+CM] - U[PI+CP]) -\|r[PI+Cl] + Wl*U[PI+Cl] )', END; By placing CI in ACARO,, CM in ACAR1, CP in ACAR2, H in $00, V in $D1, W in $D2, Wl in $D3 and PI in $X (21) could be translated into ASK as outlined in Table 2. 2.3.^ SOR using Odd-Even Storage Now we will look at another storage scheme to see if we can improve the data access time. Given an m by n mesh divide each row into k + 1 segments such that k segments are 128 mesh points long and the last segment is t points long, where < & < 128. Use two adjacent lines of PEM to store each segment. Store all the red (black) points of the segment in the first (second) line of PEM. The segments appear in the same relative order as they appeared in Straight Storage. This storage scheme will be called Odd-Even Storage. For an example see Figure 3. Appendix A explains the notation and timing method used in Table h. A point that appears in an odd (even) numbered column is considered a red (black) point. 21 If H < 6k, then Straight Storage requires m brj lines of PEM while Odd- Even Storage required m ( h£-| +l) lines of PEM. If £ > 6k then both Straight Storage and Odd-Even Storage require m VI lines of PEM. PEM Row U. Row U J+l Row U j+K-2 Row U. J+K-l h,2 u j,n-.0+lj "^n-i+d PEM 1 ! : 1 j ,3 V. ,h 1 i i i ! i lu. >n- -i+3 U. ,n- ■£+k 1 i * 1 PEM I U j,n-l PEM 63 • u. ,127 u. ,128 ; 1 • Figure 3- Odd-Even Storage Data Allocation of Row j of U Where n is Even. pi = k where k is an integer such that p < k < p + 1. 22 INITIAL CONDITIONS Register Contents Register Contents $ X PI $ D3 Wl $ DO H $ CO CI $ Dl V $ CI CM $ D2 W $ C2 CP STEP NO. ASSIGNMENT STATEMENTS OPERATIONS AND OVERLAP TIME IK CLOCKS 1 2 $A: = U[$X] (0) $C3: = $D3; $A: = = $A*$C3 PE fetch CU fetch Real multiplication 7 10 3 $S: = $A - \|f[$X] (o) (PE fetch overlapped with mult.) Real subtraction 7 h $C3: = $C0 + 1 (CU operations overlapped) 5 $R: = RTL (1,,U(3)) (PE fetch overlapped by addition) Route 3 6 $A: = $R + U[$X](3) (PE fetch partially overlapped by Route) Real addition 11 7 $C3: = $D1; $A: = = $A*$C3 (CU fetch overlapped by addition) Real multiplication 9 8 $R: = U[$X](1) (PE fetch overlapped by multiplication' 9 $S: = $A + $S Real addition 7 10 $A: = $U[$X] (2) + $R (PE fetch overlapped by addition) Real addition 7 11 $C3: = $D0; $A: = = $A*$C3 (CU fetch overlapped by addition) Real multiplication 9 12 $A: = $A + $S Real addition 7 13 $C3: = $D2; $A: = = $A^$C3 (CU fetch overlapped by addition) Real multiplication 9 Ik U[$X] (0): = $A PE store 7 TOTAL TIME 93 Table 3- Timing of an Execution of Phase 1 SOR using Odd-Even Storage 23 Phase 1 of the implementation of SOR using Odd-Even Storage requires only one PE index and one route. We can access the data using PI = (2,0,0,0. . ,0) and CI equal the first row of PEM containing mesh points to be improved during this execution. We need two assignment statements: (22.1) for the red points, and (22.2) for the black points. Using K, W, V, H and Wl are defined in 2.2-3, SOR using Odd-Even Storage can be written in GLYPNIR as follows : LOOP CI: = C, K, CK DO BEGIN CM: = CI - K; CP: = CI + K; U[PI + CI]: = W*(V*(R1R(1,, U[CI+1]) + U[PI + CI + l]) + H*(U[PI + CM] + U[PI + CP]) - V[PI + CI] + WI*U[PI + CI]); (22.1) U[CT]: = W*(V*(RTL(1,,U[PI + CI - l]) + U[CI - l]) + H*(U[CM] + U[CP])- V[Cl] + W1*V[CI]); (22.2) END; Table 3 suggests how (22.1) could be coded in ASK. (22.2) requires different code but the logic is similar. The code for SOR using Odd -Even Storage is 10$ faster than the code for SOR using Straight Storage. This is due to a savings in memory fetching and the elimination of one route. 2.3-5 SOR Machine Efficiency If a mesh has m rows and 32 {6k) columns and straight (odd-even) storage is used, machine efficiency can be doubled by placing rows 1 to - in PE's to 31 and rows — + 1 to m in PE's 31 to 63. This improved storage scheme requires that rows ^- and r-+ 1 be handled as special cases. The idea behind this technique can be used on any mesh which doesn't have a multiple 2U of 6U (128) columns when straight (odd-even) storage is used. Implementation of such techniques increases machine efficiency and decreases storage require- ments per mesh, but also increases the complexity of the program. 2.U Implementation of SLOR on ILLIAC IV 2.U.1 Parallelism in SLOR In 2.3.2 the scheme for SOR was altered to maximize parallelism on ILLIAC IV. In a similar manner, we will modify the scheme for SLOR to maxi- mize the number of columns which can be improved simultaneously. In 2.1*. 2 the need for blocks of 128 columns to perform column relaxation with optimal efficiency on ILLIAC IV will be explained. This leaves no restrictions on the number of rows other than the limits of available core. Analogous to the SOR scheme, note that column i can be improved for the second time after column i + 1 is improved for the first time. By taking advantage of this fact we can improve half of the columns simultaneously after a finite number of iterations. Thus once we reach that state we can improve the entire mesh in two steps if we have enough PEs. In SOR we improve the odd (even) points and then the even (odd) points. In SLOR we must improve the odd (even) columns and then the even (odd) columns. The proof we used in 2.1.2 to show that we could improve the points in SOR in any order can be extended to show that the order in which the columns are improved SLOR does not change the asymptotic rate of convergence. U (Ul) = h(U ( ^» ♦ U<\ +1 > ) ♦ 0>W> + D «> ). ug f . . (23.1) 1»J 1+1 »J 1-1 »J 1,J + 1 l,j-l l.J / X / (*) / (M (M u + (1 - w (u: : - h (u j . + u ; j i»j 1+1, 3 1-1, j for the odd columns and u: for the even columns - htu!** 1 ! ♦ US 4 ! 1 ' ) ♦ «r (u! U i» ♦ o!^)- -8 *. . (23.2) 1-1, j i,j+i i,j-l' i,j (A) , (i) (I) : : - hu J , + u : ! , 1 j 1+1, j i--i».i (1 _ u>) (u: : - h(u: .; , + u: 1 J) 25 2.U.2 Storage for SLOR Now we need to find a storage scheme which is efficient on ILLIAC IV. If we use Straight Storage, we run into the same inefficiency we did with that storage in implementing SOR. We need to use three PE integer indiees and do two routes per mesh point. Furthermore, the method works on blocks of 256 columns. Column J and column J + 1 cannot he improved simultaneously, Thus we improve the odd (even) columns in the set {J|I < J < I + 63} and the even (odd) columns in the set {j|l + 128 < J <, I + 191). If we use Odd-Even Storage we can eliminate one route and two of the PE indices. We also decrease the blocks to 128 columns each. 2.U.3 Improving a Line by the Cut hill -Varga Method The method of implementing SLOR must solve a tridiagonal matrix equation in each PE. Thomas' method [Todd, Page 395] is commonly used. A method by Cuthill and Varga can eliminate some of the work by doing some preconditioning if the tridiagonal matrix is positive definite and real symmetric. Even if the mesh spacings are not constant, Poisson's equation satisfies those conditions [Cuthill and Varga, Page 2l+l]. We will apply the latter method, the Cuthill -Varga method, as follows* Because we are performing successive column over-relaxation we will use a permutation of A, PAP , to discuss the technique. The equation under consideration will be -PAP~ PU = -PM. 26 -PAP -1 B, C ■■ C, B, C C, B, C XV s - \ "» \ C, B, C C, B, c c, B - (2k) where B = e, d d, e, d v d, e, d d, e C = — I, e = +2 l^+ —■), d = ~? h 2 h 2 h 2 h x y \ y x / .th B and C are m-2 by m-2 matrices. Let U. and NL denote the i column of PU and -PM respectively. Now using (2k) we obtain BU. = M. - C(U. . +-U. _ ). l i l+l l-l (25) By assuming the left side of (25) is constant we are left with a tridiagonal system of equations. Because B satisfies the requirements for the Cuthill- Varga method, [Cuthill and Varga, Page 237] > we can define a diagonal matrix G = 'm-2 J such that G -""BG -1 = T'T, 27 where T = 1 t 1 1 t. X X 1 I r 3 The elements of G and T are calculated as follows: 1 d 2 2 1 c = e 2 ; c = {e - (- ) } , 2 < j < m-2, 3 j-1 and (26) tj = C J Vl 1 < J < m-3. (27) By multiplying BU. = M. - C(U. . + U. _ ) l l l+l l-l by G we obtain G 1 BG~ 1 GU. = G _1 M. - G 1 CG' 1 G(U.^ n + U. , ) . l l l+l l-l Substituting Y. for GU. gives G^BG^Y. = G _1 M. - G^CG" 1 (Y. +Y ) i i l+l i-1 or T TY.= G'^M. - G _1 CG _1 (Y + Y ) l l l+l l-l (28) Equation (28) is first solved for TY. and then for Y . . i l The Cuthill-Varga method employs a two part algorithm to perform SLOR, First Y. is calculated usim l T TY 1 1+1 1-1 (29.1) if i is odd and * T equals the transpose of T. 28 T'x?. ( * +1) = G"V G^CG- 1 (t[H 1] + Y<_\ +1) ) (29.2) 'j if i is even. Then y is combined with Y. to perform the over-relaxation, i i (ML) .„ jjU+l) _,<*>] +Y U) _ (30) i i i i After all the iterations are completed, Y. is converted to U using G _1 Y. = U.. l i 2.U.U SLOR Implementation Nov let us discuss more specifically how we implement SLOR to solve — ^ + — — = y. Odd-Even Storage issued for the reasons stated in 6X 6Y ; Section 2.1*. 2. We use the Cuthill-Varga method and thus do the following preconditioning: calculate c, 1 £ i < m-2; t., 1 < i ^ m-3; — , 1 < i < m-2; i i and — , 1 < i < m-2. We use PEM storage for these values so that when (h c.) 2 y i we calculate c. we can do 6k of the square roots simultaneously because the square root takes a great deal of time relative to other operations on ** 1 —1 ILLIAC IV. We use Grabone ' to access the c, t., -r— and =— • We 1 x i f-u \ 2 x (h c i ) subtract the row boundary conditions times — to the appropriate row _ h d ~_L x of f and then we multiply by G . We multiply column boundary conditions by G to obtain Y and Y so that they may be used in equations (29) and (30). Now we are ready to do the iterations.*** Two phases are used. Phase 1 improves 6U columns simultaneously and Phase 2 improves the remainder. Phase 1 is used when the number of interior columns is greater than or equal to 128. Phase 1 starts at the left and works on groups of 128 interior col urn We limit our program to handle an even number of columns. The GLYPNIR function is used to access a single PE value and send it to all PEs. Refer to Table 9. ### The reader should be familiar with the material in Appendix A. 29 moving to the right -until the number of remaining interior columns is less than 128. Since there are elements of 63 interior odd columns on the first line of PEM containing interior points we use PI and CI as defined in 2.3.^. *-2 rows of temporary storage, STORE, are used. We use two similar codes for Phase 1: one for the odd columns and one for the even columns. Phase 2 can use the same code as Phase 1 if the mode is changed at the appropriate time. We will limit our discussion to the code for the odd columns, Phase 1. The rest of the code can be found in Appendix D. ST0RE[0]:= t[PI+Cl]-GRABONE(DISB[0],0)*(RTR(l, ,U[CI+l] )+U[PI+CI+l] ); LOOP CJ:= 1,1, CM3 DO BEGIN C: = CJ.[ 16:^2]; CI: = CI+K; (31) STORE [CJ]:= ^[PI+Cl]-GRABONE(DISB[C] ,CJ)*(RTR(l, ,U[CI+l] )+ U[PI+CI+1])-GMB0NE(E[C],CJ)*ST0RE[CJ-1]; END; LOOP MC:=1,1,CM3 DO BEGIN C J : =CR3-MC ; C:=CJ.[l6:If2]; ST0RE[CJ]:=ST0RE[CJ]-GRAB0NE(E[C],CJ)*ST0RE[CJ+1]; (32) END; * -1 -1 DISB contains — , 4* contains G M and E contains t.. 30 trtfTTAL c i 6nT)TtTon's' Fflfi J?'fll^WARL) elimination Register $A $X Contents STORE I C J- 1 J PI STEP NO. ASSIGNMENT STATEMENTS GRAB0NE(EICJ,CU) $B $A $S = $A*$B = * [$X](0)-J $R: = RTR(1„U(2)) $A: = U[$X](2)+$R $R $A $A = GRABONE(DISB[C],CU) = $A^$R = $S-$A STORE (I): = $A Register fco $C1 $C2 Contents CI CJ CI+1 OPERATIONS AND OVERLAP TIME IN CLOCKS Load 10 Real multiplication 9 (PE fetch overlapped by mult. ) j 7 Real subtraction (PE fetch overlapped by subtract.) 3 Route (PE fetch partially overlapped by route) Real addition 11 Load 10 Real multiplication 9 Real subtraction 7 PE store I 7 Subtotal 7T INITIAL CONDITIONS FOR BACKWARD ELIMINATION Register Contents ST0RELCJ+1J Register "fccT Contents -$A~ CJ STEP NO. ! ASSIGNMENT STATEMENTS OPERATIONS AND OVERLAP [TIME IN 'CLOCKS $B $A $A = GRABONE(E[C],CJ) = $A*$B = STORE (0)-$A STORE(O): = $A Load 10 Real multiplication ! 9 i (PE fetch overlapped by mult. ) j 7 Real subtraction ■ PE store 7 Subtotal 33 INITIAL CONDITIONS FOR OVER -RELAXATION Register Contents Register Contents $X $C0 PI CI $C1 $C2 CU W STEP NO. ASSIGNMENT STATEMENTS $R: $A: $A: UI$XJ(0) STORE (l)-$R $C2*$A OPERATIONS AND OVERLAP ,TIME IN CLOCKS $A: = $A+$R U[$X](0): = $A PE fetch PE fetch Real subtraction (CU fetch overlapped by sub. ) Real multiplication Real addition PE store Ik 9 7 Subtotal Th~ TOTAL TIME 150 Table k. Timing of an Execution of Phase 1 SLOR 31 LOOP CJ:=0,1,CM3 DO BEGIN ST:=U[PI+Cl]; Tj[PI+Cl]:=W*(STORE[CJ]-U[PI+Cl])+U[PI+Cl]; IF ABS(ST-U[PI+CI]) GRT BOUND THEN B: =1; % CHECKING THE ERROR BOUND (33) CI : =CI+K; END; SLOR has three primary sections: (31) does the forward elimination, (32) does the backward elimination, and (33) performs the over-relaxation. In Table k- we suggest how to code these sections efficiently in ASK and we give a time estimate for that code. 2.5 Implementation of API on ILLIAC IV ADI improves all columns simultaneously and then improves all rows simultaneously. Thus, when we do the column (row) improvement we want the values of the boundary columns (rows), but the boundary rows (columns) could have been added to vector t in preconditioning. One solution is to include both the row and the column boundary values in each iteration. Another solution is to add them both in when preconditioning and then set the boundary values to zero so they will not affect the interior points when used in the iteration. If we need the values of the boundary after the problem has converged then we must store them before performing any of the iterations and replace them after all the iterations are completed. 32 2.5.1 Thomas' Method on ILLIAC IV As in SLOR we have sets of tridiagonal matrices to solve. We used the Cuthill-Varga method in SLOR. This method does preconditioning to cut down on the computational time (see Section 2.U.3). ADI changes the diagonal at every iteration. Thus the preconditioning would have to be done at each interation. Thomas' Method [Todd, page 395] has the. same problem but the preconditioning requires less computer time and the total computer time per iteration is smaller. Thus we will use Thomas' Method in solving the tridiagonal matrix problems. Instead of implementing the method in a straight forward manner, we use the fact that the horizontal coefficients and the vertical coefficients are constant. The matrix problem* we want to solve is of the form aT l + b T 2 = °1 bT ± _ ± + aT. + bT. +1 = D. (1 = 2, 3,-..., n-3), bT 5 +aT Q =D • n-3 n-2 n-2 where ,2 . 1 a v~2~ + W £v > "b = — for row relaxation y h and ,2 « 1 a. - -[ 2 + w^j, b = — for column relaxation x h x Thomas' method is as follows e l=h \ = a-bV n (i =2, 3/ .,., n-3), (3k) l-l D x D. bq. q l = —> q i = a-b"e X " (i =2, 3, ..., n-2), (35) i-1 T n-2 = V 2 > T i = q i - e i T i+ i (i - n "3, *4, -.., I)- (36) N We are considering an m by n mesh, U, 33 If we let c = - we can rewrite the equation (35) as follows: b L x = e ] _ c D v q. = e. (c D. - q.^) (i = 2,3, ..., n-2), and e = ___J? . This reduces the number of divisions which are re- 3 n-3 n-2 " a-b e latively time consuming on ILLIAC IV. We have to calculate the e.'s and store them. For each row (column), they are the same. Calculating the e.'s is a serial process. The only way we could overlap is to calculate the set of e.'s for different itera- tions simultaneously. The problem, with this solution is that it requires extra storage. If we calculate all the e.'s needed for 32 iterations we use max (m-2, n-2), rows of storage. If storage is available, then this is a feasible method; but if we must use disk I/O, the computational time saved "will be overcome by increased I/O time. 2.5.2 Skewed Storage The next problem we must overcome is how to access the rows on one sweep and then the columns on the next. Skewed storage was designed just for that purpose. We can place a m by n mesh in skewed Storage by performing the following algorithm: 1) Calculate the number of rows of PEM required to store n points. Set K equal to that number. 2) Place the m by n mesh in straight storage. 3) Route each of the K rows of PEM containing elements of mesh row i. Let the distance of the route be i-1. 3U Row U 1 Row U +1 Row U. PEM PEM 1 [ t \ ' U 1,1 u u. Row U. 2+1 Row U 33 Row U +1 1,65 2,64 U 12 iu 1,66 | ! U 2,i I !u. 2,65 ! ^33, 33 1 | U 33,34 Row U 34 Row U , +1 U u Row U 96 Row U-^+l 96 3^,32! 34,96; u, 96,34 U 1,32 u 1,96 !u. 2,31 u, 2,95 u 33,64 ^2,32 J U 2,33| U. ! U 34,63i u, 96,1 96,65 u 1,33 U 1,34 •, y^ s U 33,1 u 33,65' U 34,64 u 96,2 u 34,1! u 34,65 v. 96,66 u. 96,3 u. 96,67 PEM 31 PEM 32 PEM . 33 • • • PEM 62 PEM 63 U 1,63 u. 2,62 u 33,31 u u 33,95| u 34,32 u 34,99 u 96,32 u 96,96 1,64 ^2,63 U 33,32 u 33,96 U 3^,31] u 34,95 u 96,33 Figure 4 . A 96 by 96 Mesh in Skewed Storage 35 AD I is a simultaneous method, (i.e. only values from the previous iteration are used, see (l5.l) and (15.2)). On ILLIAC IV v e can simul- taneously improve up to 6k rows (columns). Thus if the dimensions of the mesh are less than 67 we can do all the rows (columns) in one time step. If one of the dimensions is greater than 66 then we need to use more than one time step for row or column relaxation. In this case we will use two phases to complete an iteration. Anm by n mesh will be divided using R m-2 and J = j-rj— • Phase 1 row (column) relaxation will improve R (j) groups of 6k consecutive rows (columns). Then Phase 2 will improve the remaining interior rows (columns). Phase 1 starts with row (column) 2 and works toward row (column) m-1 (n -l)- When Phase 1 im- th proves the I group where 1 < I < R (j) and k is the first row (column) of group I, the old values of row (column) k-1 are needed. Thus to im- prove group I Phase 1 goes through the following steps: 1) The values of row (column) k-1 are placed in the temporary storage , NEW. 2) The values in the temporary storage, OLD are placed in the PEM storage for row (column) k-1. 3) The values or row (column) k+63 are placed in the temporary storage, OLD. k) Rows (columns) k through k+63 are improved. 5) The values in NEW are placed in row (column) k-1. When Phase 1 is improving the first group steps 1, 2 and 5 can be omitted. Phase 2 needs to do steps 1, 2, k and 5. 36 Now we will discuss the type of indexing used in accessing the mesh points required to perform Phase I column relaxation using skewed storage. We ac- cess the set {(\|r. ., U. . )|k < j < k + 63} by using a permutation of 1 j 3 1, 3 the PE integer, (1,0,0, •••,0). PI will stand for that permutation. PI will be used as a PE index and CI will be the CU index. PI must be routed one PE to the right to access the set {U. , . |k < j < k+63}. PI l+J-j 3 must be routed one PE to the left to access [U. . |k < j < k+63} • To l - 1,3 access the set {U. . |k < j < k+63} we must add ETR(l,,Pl) to PI. The 1, 3"*" 1 set {U. . - jk < 2 < k+63} can be accessed using only CI. After being accessed the four neighbors of U. . must be routed to the PE containing 1,3 U. . before U. . can be improved. The following GLYPNTR code uses this 1,3 i>3 method of data access to perform Phase 1, Step k of column relaxation: LOOP C:=l,l,N-2 DO BEGIN CC:= C. [l6:^2]; CI:=CI+K; U[PI+CI]: = (AI*0 [PI+CI]-BETA*(RTL(1,,U[PI+RPI+CI])+RTR(1, ,u[ci])) +CA*U[PI+Cl])-RTR(l,,U[LPI+CI-K]))*GRABONE(B[C-l],CC); (37)" LPI : =PI ; PI : =RPI ; RPI : =RTR ( 1, , RPI ) ; END; LOOP IC:=0,l,N-3 DO BEGIN C:=N-2-IC; CC:+C.[l6:U2]; CI:=CI-K; U[PI+CI]:=U[PI+CI]-GRAB0NE(B[C-1],CC)*RTL(1,,U[RPI+CI+K]); (38) RPI:=PI; PI:=RTL(1, ,PI);END; * The variable, k equals the first column of the group being improved. 2 2 ** AI=-h ; BETA= l/h ;W equals the relaxation parameter for this iteration; CA=2*BETA+W. 37 The code is divided into two sections: Section (37) does Thomas' forward elimination. Section (38) does Thomas' backward elimination; Table 7 suggests how this code could be translated into ASK and how much time the ASK code would take. By replacing PI with KPEN=ETR(l, ,PENK) where PENK=(0,K,2k, . . . ,63K) and K is the number of lines of PEM required to store a row of the mesh, we would have Phase 1 row relaxation.* A complete program for a m by n mesh where n, m < 64 is found in Appendix E. * In row relaxation a PE index must be used to access {U. . , k < j < k+63} 38 INITIAL UWWlTiWIlJ WW HJ1WA1W kliMlMM r Register Contents STEP NO 8 9 10 11 12 13 Ik 15 16 17 18 19 20 21 22 23 fcCl $D0 $D1 CI AI BETA ASSIGNMENT STATEMENT $A: = U[$X](1) $C2: = $D2;$A:=$A*$C2 $S: - $A + *[$X](1) PI $X $B = $X = $X + RPI = RTL(l,,U[$X](l)) $A: = RTR(l,,U(l)) $X $A $R $C2 - LPI = $A + $B = U[$X](3) : = $D1;$A:=$A*$C2 $X: = PI $A: = $S-$A LPI: = $X $C2: --= $DQ;$A:=$A*$C2 $B $A $S $R $A RPI = RTR(l,,$R) = $A+$B = RPT = RTR(1,,$S) =» $A*GRABONE(B[CI-l],CC) : ■ $R U[$X](1): = $A $X: = $S Register Contents 6D2 fcx CA PI CI-K OPERATIONS AND OVERLAP TIME IN CLOCKS PE fetch 7 CU fetch and real mult. 10 (PE fetch overlapped by mult.) Real addition (PE store overlapped by add. ) PE fetch and integer addition (PE fetch partially overlapped by addition) Route (PE fetch partially overlapped by route) Route 7 (PE fetch partially overlapped k Real addition 7 (PE fetch overlapped by add. ) CU fetch, real multiplication 10 (PE fetch overlapped by mult.) Real subtraction 7 (PE store overlapped by subt.) CU fetch, real multiplication 10 Route 3 Real addition 7 (PE fetch overlapped by add. ) Route 3 Load real multiplication 19 (PE sbore overlapped by mult.) PE store 7 PE register to PE register load 1 126" Subtotal INITIAL CONDITIONS FOR BACKWARD ELIMINATION Register Contents Register •Contents Register Contents JF PI $A U[RPI+CI+KJ feci CI STEP NO, ASSIGNMENT STATEMENT OPERATIONS AND OVERLAP TIME IN CLOCKS $A $A = GRAB0NE(B[C-1],CC) = $B*$RTL(1,,$A) = U[$X](l)-$A U[$X](1): = $A $X: = RTL(1, ,$X) Load 10 Route, real multiplication 12 (PE fetch overlapped by mult.) Real subtraction 7 PE store 7 (Route overlapped by store) Subtotal lo" TOTAL TIME TST Table 5. Timing of an Execution of Phase 1 ADI 39 2.6 Summary of the Results for Iterative Methods Now that we have examined implementation of the various methods we can make some recommendations concerning the use of the methods. Table 6 contains a numerical study of convergence rates of the method applied to Poisson's equation where h = h = x y 15 error bound was 0.0001. on different mesh sizes. The GhxGk 32x61+ 32x32 16x61+ i^^^° 1 1 1. 86415 1. 79300 1.71+760 1. 6I+750 tt) O t" 1 k 63 1+0 32 21 HMO p g td P3 5 ii i a 66 hi | 33 23 ct- H- OJ 1. 9061+5 1.85610 1.82147 H 1.75050 ra h 92 59 kl 32 to i a 90 63 kk 28 I a 10 9 8 7 6 H 1 9 8 7 6 a) co is the relaxation parameter b) I is the theoretical number of iterations c) I is the actual number of iterations* a d) 2ri is the number of relaxation parameters used in ADI Table 6. A Numerical Study of the Methods under Discussion We can see from Table 6 that ADI is consistently the fastest** then comes SL0R, with S0R being slowest. This conforms with theoretical results. In fact, in a square mesh where h = h , SL0R converges /2 times faster than x y S0R while ADI converges B times faster than S0R. B is a monotonically increasing function of the number of mesh points [Todd, Pages 396-398]. #* Remember ADI improves the points twice in each iteration. The speed of the algorithms is compared with respect to the number of iterations . ko To compare the time required to perform one of the above methods on ILLIAC IV to that required to perform the method on a specific machine -we need to calculate the clocks per point per iteration and compare it with the clocks per point per iteration on ILLIAC IV. We have shown that SOR, SLOR, and ADI can be programmed to enable ILLIAC IV to improve 64 points simulta- neously for certain mesh sizes. By dividing the clocks required to improve 64 points simultaneously by 61+ we obtain the clocks per point per iteration on ILLIAC IV. Table 7 contains these values and other pertinent information. ADI SLOR S0R Number of operations per point per iteration Mult. 10 k k Add. 10 6 5 Time in clocks per 6k points per iteration Arithmetic 160 78 71 Data Access 164 72 22 Total clocks per point per iteration 5.o6 2.34 1.45 Optimum mesh size** 641 by 64L 1281 by L 641 by 2L Rows of PEM required for temporary Max (n-2,m-2) m-2 +, fo-2) e storage h 6k Table 7. Summary of Time and Storage Requirements for the Iterative Methods on a m by n Mesh. Most of the data access time on ILLIAC IV could be eliminated if it was a serial machine. Let Machine A be a serial machine with a processor similar in speed to a PE but with the ability to perform ADI, SOR, and SLOR with neg- ligible data access time. From Table 7 we can see that Machine A would tab 71/1.45, about 48, times longer to perform SOR ' than ILLIAC IV would take. Machine A would take about 32 times longer than ILLIAC IV to perform ADI or SLOR. For maximum efficiency on ILLIAC IV, one dimension of the mesh must be a multiple of 6k for SOR and SLOR while both dimensions must be multiples of 6k for ADI. ** I and L are positive integers A single PE has approximately twice the arithmetic speed of a CDC 6600. #### The clocks per point per iteration on Machine A equal the arithmetic time in clocks per 6k points per iteration on ILLIAC IV. 1+1 The times in Table 7 are assuming the inner loops of the programs are written in ASK. If the programs were written completely in GLYPNIR the times would about double. Tables 6 and 7 indicate that ADI is the fastest iterative algorithm on ILLIAC IV for rectangular meshes. The theory for ADI has only been developed for special cases [Young, Page 555]. If we have a commutative case, HV = VH, then ADI will be effective. The commutative space requires a rectangular region and coefficients of the differential equation which are sufficiently regular [Young, Page 538]. In some non commutative cases, numerical experiments have shown ADI to converge rapidly. This is not always true. In some non commutative cases ADI fails to converge for cer- tain parameters [Young, Page 5^6], More work needs to be done on non- commutative cases before ADI can be used freely on them. ADI performs row relaxation followed by column relaxation. Thus to efficiently perform ADI on a m by n mesh on ILLIAC IV, both m and n must be divisible by 6k. For example if we have a l6 by 6k mesh when we per- form row relaxation, ILLIAC IV will be working at 25% machine efficiency while for column relaxation ILLIAC IV would be working at 100% machine efficiency. If one had four 16 by 6k meshes to solve, ILLIAC IV could solve the four meshes simultaneously and thus brings the machine effi- I ciency up to 100%. SLOR converges in fewer iterations than SOR and can be programmed to take about the same amount of time per iteration as SOR on most serial machines. On ILLIAC IV SLOR requires data from one PE broadcast to all the PEs while SOR does not. This is the major factor in causing SLOR to U2 take about 1.5 times longer per iteration than SOR, see Table 7. Thus SOE is a faster method on ILLIAC IV unless it takes at least 1.5 times more iterations to converge. This is rarely the case. SLOR requires considerably more temporary storage than SOR. Finally SLOR is a block iterative method and thus cannot be as easily reformulated as SOR for efficient performance on ILLIAC IV. As shown in Section 2.3 of this study, SOR has a few constraints on how it must be implemented. This enables one to program it effi- ciently on ILLIAC IV for most mesh geometries. SOR seems to be the most promising of the iterative and direct methods examined by the author for non-rectangular meshes. In this study, we indicated that the order in which the elements are improved by SOR does not affect the asymptotic rate of convergence. This can be misunderstood. The number of iterations required to obtain a specified error bound is dependent on the order in which the elements are improved. For specific initial conditions, one ordering might give a faster initial rate of convergence than another. In some applications of Pois son's equation the actual number of iterations required for con- vergence is substantially smaller than theoretically expected because a good initial guess is supplied. In these cases the ordering has a greater affect on the computer time required to get an acceptable solution. ADI and SLOR require a considerable amount of preconditioning before the iterative process can begin. Thus if the initial guess is good enough, SOR will take the least amount of computer time on any mesh. 1*3 This study examined straight and odd-even storage for SOR. The use of odd-even storage improved the speed of SOR by 10%. In most problems, straight storage is used. Thus, for a valid comparison of the two storage schemes, the time to convert straight storage to odd- even storage must be considered. This takes about the same amount of time as one SOR iteration. Thus the program must require at least 20 iterations to justify converting to odd-even storage and then back to straight storage. uu 3. IMPLEMENTATION OF DIRECT METHODS ON ILLIAC IV 3.1 Introduction to Hockney's Direct Method (FACR ) Direct methods for solution of a restricted class of Poisson's equation have "been developed which are faster than any iterative method developed to date [Dorr, Pages 258-259]. To the author's knowledge, Hockney's Fourier Analysis /Cyclic Reduction (FACR) is the fastest direct method on serial machines [Hockney, Page 159]. In this paper we will examine FACR and modify it so that it can be programmed efficiently on ILLIAC IV. FACR solves the "five-point" difference formula on a rectangular mesh; namely, u i + -2U . + U ._ . U , _ -2U , + U , ._ s-l,t s,t s+l,t s,t-I ■ S,t S,t+1 „, / v ^2 ,2 " s,t Uy; h h ' x y N where the number of points being computed in one direction is 2 -1 for N Dirichlet's boundary conditions, 2 for periodic boundary conditions and N 2 +1 for Neumann's boundary conditions. These three boundary conditions are permitted in the x and y direction, giving nine possible combinations of boundary conditions. A 2 by 2 mesh produces a linear system with 2 unknowns. Let M = 2 and N = 2 . FACR solves this system using the following five step algorithm: 1. Given y compute ¥ for the even columns and overwrite ¥ with ¥ on the even columns . 2. Using T compute ¥ and overwrite ¥ with ¥ . 3. Using ¥ solve for U and overwrite ¥ with U . i s s U. Using U compute U on the even columns and overwrite U with U on the even columns. h5 5. Using the values of U on the even columns solve (39) for the values of U on the odd columns and overwrite ¥ with U on the odd columns. The superscripts are for Dirichlet's "boundary conditions and are defined as follows: ur = w _ i,J i,j+l M-l 2 r ...* . Trik f i,j = B J, v l) Sln — 2 V„ , ,ik U i,J = M I U k,J k=l * J Odd/even reduction divides the problem into two parts; first to solve a linear system concerning even columns and secondly to solve for the values on the odd columns. The even columns form a linear system with M^ unknowns which is solved by steps 2, 3, and k. Step 5 involves the r- linear systems with M unknowns that give the values for the odd columns. Fourier analysis decouples the even columns into M tridiagonal linear N s systems with — unknowns. The unknowns are the Fourier harmonics, U , of U. Recursive cyclic reduction solves the linear systems for the Fourier harmonics U . Fourier synthesis converts the Fourier harmonics U obtained by the previous step to U giving the values of U on the even columns. The values of U on the even columns are used to calculate the values N of U on the odd columns. This involves solving — tridiagonal linear systems with M unknowns [Hockney, Pages 1U8-153]. k6 Refer to Appendix B for a more detailed explanation. Note that all five ste-DS can be performed using one mesh containing the "boundary conditions and ¥ initially. This is overwritten by f , ! , U and finally the solution U. We will discuss the method in three parts, steps 1 and 5 S step 3, and steps 2 and k. Suggestions will he made for the application of each part. Finally we will present a program in GLYPNIR similar to FACR and make suggestions on how it can he programmed for ASK. 3.2 Odd/Even Reduction and Odd Column Solution Odd/even reduction is used to cut down on the number of columns where Fourier analysis and synthesis is applied. Consider the three neighboring equations: # "t-2 + BU t-l + ^ U t = Vl y y \ \ $ u t + Bu t + 1 + ^ 6k then yzT~\ > L 9 o 1 and odd/even reduction would reduce the number of times Fourier analysis and synthesis must be applied. The best case is where j tj-J = 2 pr^TT J in which case the number of Fourier analysis and synthesis is cut in half. Straight storage is the standard storage scheme used by most pro- grams. If odd/even reduction is used, maximum machine efficiency is obtained when an even column is contained in each PE for the odd/even * fPl equals the integer L where L - 1 < P < L. U8 96 x 96 •P 0) bO ro •H Sh ro o ^ -p -p CO CO PE PE 1 U E 2 • • • PE 31 PE 32 PE 33 PE 62 PE 63 s %,2 %3 p ^ ? 32 %33 %3^ ^,63 p U ; 6U | P U,66 p U ? 6 7 P U ?9 6 -- -- -- : p 5,2 P 5 ? 3 p 5,32 P5,33 P 5;3^ p 5 ? 63 p 5,6l+ p 5,66 P 5 ? 6 7 P 5;96 " -- -- ^VV\/ N A/ a/«/v ^-Wv %V^ aa..-\aa */ / > u o o o ■» CO •H W 1) -P fl >S M G M-P ro o O T5 P <1) co K C H ro o o w •H 13 o a < ro ./WV. ^Av^'\ NtV-WV V p k,2 P ^;3 p U ? 65 P U,66 P 5,l p 5 ? 2 P 5 ? 3 p 5,65 p 5,66 '■• v v . , Aaa^.-V /W\/| K,32 ^95 5,32 ^1 N rvV/j P S33 1 ^6j 3,33 ^2§J (WW "^M/v-y i p ^ p U ? 63 __ -- fc,3^ p 5 ; 63 i \f : "... •;' . ,'•-.■ V ' >, /V^' P J4 J 6U_ ,5,6^ T3 d) £ ro 0) fH ,* o CO -P CO rvVV* ^,63 ,5,62 v\ p )4,6U ^,63 I/" <■•■ \\ p i+,l p j ±2 65. ^,6U p %95 %95 5,29 ^93 V v v A N \' ^,30 J ^,31 J 3,30 !p 5^L >A A VVvVi ^ 3,31 2i25. NV P U,32 I K,6i ! \ p h,62 ^,60 AA^s Figure 5. Storage Schemes Used on Rows k and 5 of a 96 by 96 Mesh if Odd/Even Reduction is Used ; p 5,6i I ■\/wJ k9 reduction step, the Fourier analysis step and the Fourier synthesis step; CRED can access a different row in each PE; and during the solution of the odd columns an odd column can be accessed by each PE. The following storage changes will fulfill all these conditions assuming each row of the mesh requires an even number of rows of PEM. 1. We assume the main program supplies U in straight storage. 2. Before odd/even reduction, the even rows of PEM containing U are routed one PU to the right of the odd rows. 3. Before CRED, U is placed in skewed storage. h. Before Fourier synthesis, U is returned to the storage used for odd/ even reduction. 5. The mesh U is placed in straight storage at the end of the subroutine. See Figure 5 for clarification of the above storage schemes. If odd/ even reduction is not used steps 2 and h of the storage changes need not be executed. The use of odd/even reduction requires the extra storage changes because steps one through four of FACR are applied to even columns while step five solves the odd columns . Without the use of odd/even reduction steps 2, 3, and h in Section 3.1 would be applied to all the columns. 3.3 CRED In step 3, section 3.1 we have a tridiagonal system of equations to solve. The derivation of the coefficients for Dirichlet's boundary conditions is contained in Appendix B. By multiplying equation (B-8) in 2 th Appendix B by h the diagonal coefficient for the k row becomes 50 -=-K^-¥-«(^^4-) -^ + 6^ +8 g +2 ; h ' h h X XX XX Hockney solves these tridiagonal systems using cyclic reduction recursively. Given lt-U + \lt-2 + lt =\ 2 ^,t-l lt-2 + \ It + lt + 2 =h y 2 lt ^ U k,t + ** U k,t + 2 + K,t + k ■■ h / *k,fl where t is even. By multiplying the middle equation by -A^and adding we obtain it-* + < 2 - # it + it+u ■ \ 2 ( \,t- 2 + \,t +2 - «;,*' (1*3) Now the process is repeated on the equations until we are left with just one equation. Then the process is reversed [Hockney, Page 150]. This method uses Log N extra words of storage, UN - 2Log p N additions, 2N multiplications and Log N divisions for an N variable system. It also requires that N equal 2 - 1 if the system has Dirichlet conditions, 2 if the system has periodic conditions, and 2 + 1 if the system has Neumann conditions. For Dirichlet boundary conditions Thomas' method allows one to solve a tridiagonal system of N variables where there are no restrictions on N, (See Section 2.5.1). In the case where the diagonal elements are equal and all the other non-zero elements equal 1, we calculate * N is the number of interior even columns in the mesh. 51 XI = IT ' S k,i = A -ej . * for i = 2,3,...,N. (i*) k ' k k,i-l "k,i = *k,i e k,i •• \,± = (T k,i- Vi-i' e i for i = 2 '3----' H - ^s) U k,H * «k.H' U k,i= 4 k,i" e k,i li + l fOT * = B - 1 '"- 2 ' ■- 1 - ik6) This method uses N words of extra storage, N divisions, 2N multipli- cations, and 3N additions. On ILLIAC IV a division requires as much time as six multiplications. Thus the calculations of the e.'s takes most of 1 the time required for Thomas' method. Since the e.'s are independent of U and V they could be calculated in preconditioning and used repeatedly by Thomas' method. Since each row has a different A, we need a different J k set of e . ' s for each row of U. Thus all the e . 's would require half K., i k , x as much storage as U if odd/even reduction is applied. Thomas' method can be applied to an arbitrary M. This not only allows the program to be more general but in some cases can save storage in placing U and ¥ in PEM. If a problem with Dirichlet or Neumann boundary conditions required the accuracy obtained by 6h by 6k meshes, applying cyclic reduction would require the meshes to be 65 by 65 while Thomas' method would accept 65 by 6k meshes. In straight storage a 65 by 65 mesh requires 130 lines of PEM while a 65 bv 6k mesh reauires 65 lines of PPM. Tf thp boundarv conditions are periodic, cyclic reduction would require a 6k by 6k mesh, and would require only 6k lines of PEM. The extra storage required to store the e . 's for Thomas' method is too great for some memory bound problems k, 1 By changing (k6) to U k,N = V U k ,N-l = \,N ' \ U k,N ; s s (W V = V1+1 " A k u k, i+ i - u k, i+2 for i = N - 2 > N - 3 > •••> 1 - 52 we eliminate the need for extra storage. We will refer to this method as modified Thomas' method. It takes N divisions, 2N multiplications , and UN additions. Table 8 summarizes the storage and computational time of cyclic reduction, Thomas' method and modified Thomas method where the divisions are done in preconditioning for Thomas' method and cyclic reduction and the divisions are not done in preconditioning for the modified Thomas' method The computational time is calculated by multiplying T s 9 and 56 clocks by the number of additions, multiplications and divisions respectively and is for 6k rows of U. The storage is in words not lines of PEM. Mesh Size Restrictions Extra Storage Requirements Time per 6k Rows in Clocks Thomas' Method with odd/ even reduction An odd number of columns Thomas ' Method NM/2 words 39N/2 Mod. Thomas' Method No words 102N/2 Thomas' Method without odd/even reduction No Restrictions Thomas ' Method NM words 39N Mod. Thomas ' Method No words 102N Cyclic Reduction N = 2 1 + 1 MLog N words 1+6N - Iklogjt Table 8. Comparison of Three Methods to Solve Tridiagonal Matrix Equations for Dirichlet's or Neumann's boundary conditions Odd/even reduction is most helpful when f^] = 2 fj|g] where k is the number of interior columns (see Section 2). The time estimates of Table 8 for Thomas' method with odd/ even reduction are only half the corresponding values for Thomas' method without odd/even reduction. This is because 53 odd/ even reduction decreases the size of the tridiagonal system to he solved hy a factor of 2. For cyclic reduction the amount of extra storage in Tahle 8 is misleading. If I > 6 then 6Um words of extra storage are required per mesh in PEM. If both U and ¥ are in PEM,then 128M words are required. This is because cyclic reduction requires 2 +1 columns for Dirichlet 's or Neumann's boundary conditions while a similar error bound could be obtained using Thomas' method and 2 points. For cyclic reduction on Neumann's boundary conditions, if I > 6 then an extra Fourier analysis and synthesis must be performed to solve the last column. This time should be divided by the number of rows and added to the time per row for cyclic reduction on Neumann's boundary conditions. If odd/ even reduction is used then the odd columns of U are calculated by solving tridiagonal systems. These systems are already restricted in size by the FFT performed on the even columns. One cyclic reduction pro- gram can be written to solve for U on the odd columns for all three boundary conditions. Thus cyclic reduction will be used to solve for U on the odd columns when odd/even reduction is employed. 3.^ Fourier Analysis and Synthesis Hockney wrote one routine which performs Fourier analysis or synthesis for any of the boundary conditions [Hockney, Page 201]. His routine assumes that there is plenty of temporary storage available. Although this is true on most large serial machines ILLIAC IV has a relatively small amount of storage compared with its speed. Thus a method which is as fast as Hockney' s Fourier routine but does not require extra storage would be 5U preferred. Cooley, the author of FFT, presents such a method in [Cooley, Page 320-323]. Cooley shows that it is possible to use the complex fast Fourier transform to obtain the sine series (required for Dirichlet boundary conditions), cosine series (required for Neumann boundary conditions) and real series (required for periodic boundary conditions). Both Hockney's and Cooley' s methods place restrictions on the interior mesh sizes: 2 1 -1 for Dirichlet, 2 for periodic and 2+1 for Neumann. Cooley' s algorithm for Dirichlet boundary conditions goes as follows . let M = 2 1 , Y(j) for j =1, ..., M-l be the coefficients of M-l .^ the Fourier sine series, b(k) = £ Y(j) sin „ . J=l Y(0) = Y(M) = 0, and Y(j) = - Y(2M-j) = - Y(-j) for j = 0, . . . , M. Define X(j) = -[Y(2j+1) - Y(2j-1)] + Y(2j)i for J = 0, ..., M/2. Thus X(0) = ~[Y(1) + Y(-l)] + Y(0) = -2Y(l) + Oi and X(M/2) = ~[Y(M+l) - Y(M-l)] + Y(M)i = 2Y(M-l) + Oi. For j =1, ..., M/2-1, X(M/2+j) = - Y(M+2j+l) + Y(M+2j-l) + Y(M+2j )i = Y(M-2j-l) - Y(M-2j+l) - Y(M-2j)i. Let X(j ) = C(j) and calculate A^j) and A g (j ) for J = 0, 1, . . . , M/k using VJ) = °(J) + C(M/2+j) andA 2 (j) = [c(j) - C(M/2+j)]W^ where W M = C0S ¥" + ± SIN ¥" • Then calculate A(j) and A(M/U+j ) for j = 0, 1, . . . , M/U-l 55 vhere A(j) = A x (j) + i A 2 (j), A(MA+j) = A ± (j) - iA 2 (j). Thus, for J = 0, 1, ..., M/U, A(j) = -Y(2j+1) + Y(2j-1) + Y(M-2j-l) - Y(M-2j+l) + SIN ^_ [Y(2j+l) - Y(2j-1) + Y(M-2j-l) - Y(M-2j+l)] + COS ^L [Y(2J ) + Y(M-2j)] + i{Y(M-2j) - Y(2j) + SIN ^li(Y(2j) + Y(M-2j ) ] + (1+8.1) COS —A- [Y(2j-1) - Y(2j+1) - Y(M-2j-l) + Y(M-2j+l)]} and for J +1, . . . , M/k A(M/2-j) = Y(2j-1) - Y(2j+1) - Y(M-2j+l) + Y(M-2j-l) - SIN 2gl [Y(2j+1) - Y(2j-1) + Y(M-2j-l) - Y(M-2J+l)] - COS ^- [Y(2j) + Y(M-2j)] - i {Y(M-2j) - Y(2j) - SIN ^L [ Y (2j) + Y(M-2j)] - COS ^f- [Y(2j-l) - (U8.2) ^- [Y(2j) + Y(M-2j)] - COS & Y(2j+1) - Y(M-2j-l) + Y(M-2j+l)]}. Now the complex fast Fourier transform is applied to A giving N/2-1 T jk X(j) = I A(k)w;L for j = 0, 1, ..., M/2-1. k=0 For j ' 1, 3, 5, ..., M-3 let b(j) = X (p~) imaginary and b(j+l) = X (^) real; b(m-l) = X(|-l) imaginary. Finally b(j) for j - 1, ..., M-l is calculated Ub(j) = [b(j) - b(M-j)] - [b(j)+b(M-j)]/[2Sin *£], (U 9 ) M-l Now we have Ub(j) = k £ Y(k) Sin ™- . k=l Fourier analysis calculates — b(j) and Fourier synthesis calculates b(j). Since we are dealing with linear equations we can multiply p in (l) and 56 INITIAL CONDITIONS REGISTER CONTENTS REGISTER CONTENTS REGISTER CONTENTS REGISTER CONTENTS fx PI ' $CO I $D0 J $D3 KP $R P[$X](2) $C2 I+ILJ $D1 L $D4 KM $S pr$x](3) $C3 J-I+ILJ $D2 K $D 5 ILJ LOOP ASSIGNMENT STATEMENTS OPERATIONS AND OVERLAP TIME IN CLOCKS TOTAL TIME FOR LOOP 1 P[$X](3):=$R-$S PEM Store, Addition llf 1 $C3:=$C3-$D2 CU Addition overlapped 1 $B:=P[$X](2) PE Fetch overlapped 1 $A:=$Sj$S:=$B Two PE Register Transfers 2 1 P[$X](2):=$R+$A Addition, PE Store Ik 1 j 1 $C2 : =$C2+$D2 CU Addition overlapped ,M-lv TNI 1 1 $R:=P[$X](2) PE Fetch overlapped 3°^ y P[$X](2):=$H+$P Addition, PE Store Ik $C2:=$DO+$D5-$D2 CU Addition overlapped $S:=P[$X-(2) PE Fetch overlapped $C3:=$D5 CU Register Transfer overlapped P[$X](3):,-$S-$S Addition, PE Store 14 $C3 : =$C3+$D2 CU Addition overlapped $R:=P[$X](3) ^ODD3 PE Fetch overlapped P[$X](3):=-$R-$R Addition, PE Store li+ . TNT $CO:=$D3 CU register transfer overlapped te U 2 $C3 : =$C3+$D3 J $C2 : =$C3 -$D2 CU Addition overlapped 2 $R:=P[$X](3)-$R PE Fetch overlapped, Addition 7 2 $D6: =GRABONE (DS [GL] ,N2-IL) CU Load 10 2 $D7 : =GRABONE (DS [GL] , IL) CU Load 10 2 $S : =$D6*$R Multiplication 9 2 $A:=P[$X](2)*$D7 PE Fetch overlapped, Multiplication 2 $S : =$A-$S %0DD2 Addition 7 2 $R:=$R-4D7 Multiplication 9 2 $A:=P[$X](2)*$D6 PE Fetch overlapped 9 2 $R:=$A+$R %ODDl Addition 7 2 $C3:=$DO+$DU-$CO CU Addition overlapped 2 $CO:=$CO+$D3;$Cl:,$C3-$D3 CU Addition overlapped 2 P[$X](2):=P[$X](3)-P[$X](1) Two PE Fetches, One overlapped Addition, PE Store 21 2 $C2 : =$C2+$D2 ; $C1 : =$C1+$D2 CU Addition overlapped 2 ODD3:=P[$X](2) PE Fetch overlapped, PE Store 7 2 P[$X](2):4S-P[$X](1) PE Fetch, Addition, PE Store 21 2 P[$X](1):4S+P[$X](1) PE Fetch overlapped, Addition, PE Store 11+ 2 $C2:=$C2-$D2 CU Addition overlapped 2 $S:=P[$X](2) PE fetch overlapped 2 P[$X](1):4S-$R Addition, PE Store Ik 2 $B:=$R PE Register Transfer overlapped 2 $R:=ODD3 PE Fetch overlapped «^>I3 2 P[$X](2):=$S+$B Addition, PE Store Ik $C2 : =$DU+$D1 ; $C3 : =4Di++$D5 CU Addition overlapped $R:=P[$X](2) PE Fetch overlapped P[$X](2):=P[$X](3) PE Fetch, PE Store Ik *® P[$X](3):=$R+$R Addition, PE Store Ik SUBTOTAL (57M-155)f^| Table 9. Preparation of the Data for the Fast Fourier Transform* *The logic for the code to keep current values in the ADB's defined in the initial conditions is supplied in GLYPNIR, Appendix B. 57 INITIAL CONDITIONS REGISTER CONTENTS REGISTER $D2 CONTENTS REGISTER Id! CONTENTS REGISTER |D8 CONTENTS IF $DO $D1 PI J L $D3 $DU K KP KM $D6 $D7 ILJ IP Jl $D9 $D10 12 CJ IT ! = 'ASSIGNMENT STATEMENTS 'OPERATIONS AND OVERLAP TIME IN CLOCKS TOTAL TIME FOR LOOP $C2;=$D5 CU Register transfer overlapped $C3:=$C2+$D1 $S:=P[$X](2) P[$X](2):=$S+P[$X](3) P[$X](3):=$S-P[$X](3) $C2:=$C2+$D2 !CU Addition overlapped 'PE Fetch overlapped IPE Fetch, Addition, PE Store PE Fetch overlapped, Addition, PE Store CU Addition overlapped 21 lit 35 ,M+l,rN-| $C0:=$D3 ;CU Register Transfer overlapped $C2;=GRABONE( INDEX [GL],IL); JCU Load 4 o 10 10(^, $R:=P(0) P(0):=P(2) P(2):=$R $C0 ; =$C0+1 ; $C2 ; =$C2+1 jPE Fetch jPE Fetch, PE Store iPE Store JCU Addition overlapped 7 lit 7 28 <¥ffl $D11 : =GRABONE ( DS [ GL ] , N2-N11 ] *$D8 JCU Load; Multiplication $D12:=GRAB0NE(DS[GL],N11) CU Load 1 19 10 29(M-1) $C2:=$Dl++$D7+$D5 $C3 : =$C2+$D9+$D9 ; $C1 : = $C3+$C2 $R:=P[$X](3)*$D11 $S:=P[$X](1) $B:=$S#$D12 $R:=$R-$B #0DD1 $S:=$S#$D11 $A:=P[$X](3)*$D12 $S : =$A+$S #0DD2 P[$X](3):=P[$X](2)-$R P[$X](2):=P[$X](2)+$R $C2:=$C2+$D2 P[$X](1):=P[$X](2)-$S P[$X](2):=P[$X](2)+$S ;CU Addition overlapped CU Addition overlapped PE Fetch overlapped, Multiplication PE Fetch overlapped Multiplication Addition Multiplication PE Fetch overlapped, Multiplication Addition PE Fetch overlapped, Addition, PE Store PE Fetch overlapped, Addition, PE Store CU Addition overlapped PE Fetch overlapped, Addition, PE PE Fetch overlapped, Addition, PE Store 9 9 7 9 9 7 lit lit Ik lit 106(Log 2 (M)-2) Wl 53 SUBTOTAL (^(M+l)Log 2 (M)+5M-79) N 55 3^-5^ Table 10. The Complex Fast Fourier Transform* The logic for the code to keep current values in the ADB's defined in the initial conditions is supplied in GLYPNIR, Appendix B. 58 INITIAL CONDITIONS REGISTER CONTENTS REGISTER CONTENTS REGISTER CONTENTS REGISTER CONTENTS $X PI $DO J $D1 I $D2 ILJ LOOP ASSIGNMENT STATEMENT OPERATIONS AND OVERLAP TIME IN CLOCKS TOTAL TIME FOR LOOP $D3 : =GRABONE ( IS [ GL ] , 11 ) CU Load 10 10 <^) 6 $C1:=$D2+$D1;$C2:= $D0-$D1+$D2 CU Addition overlapped 6 $R:=P[$X](1) . PE Fetch overlapped 6 $B:=P[$X](2) PE Fetch 7 6 $R:=$R=$B Addition 7 6 $A:=P[$X](1) PE Fetch overlapped 6 $C3:=$D3 CU Register transfer overlapped 6 $A:=$A+$B Addition 7 6 $S:=$A*$C3 Multiplication 9 6 P[$X](1):=$R+$S Addition, PE Store ll+ 6 P[$X](2):=$S-$R Addition, PE Store Ik 58( M+l ■>& SUBTOTAL 29M flrl +29 lt|in + 5Mf5 Table 11. Final step in Cooley's Fourier Analysis and Synthesis* *The logic for the code to keep current values in the ADB's defined in the initial conditions is supplied in GLYPNIR, Appendix B. 59 INITIAL CONDITIONS ]GISTER CONTENTS REGISTER CONTENTS REGISTER CONTENTS REGISTER CONTENTS $X $D0 PI CI $D1 $D2 IL L2 $D3 $Dl+ L3 K $D5 $D6 KM1 )0P ASSIGNMENT STATEMENTS IOPERATIONS AND OVERLAP ' ™® _J N * ^^r oSf CLOCKS FOR LOOP 7 $C2:=$D1;$C3:=$D0 CU Register Transfer overlapped 7 P(2):=RTR($C3,,P(2)); PE Fetch, Route, PE Store 20 20fe-l)|^ $C1:=0;$C3:+$DO CU Register clear overlapped $S:=1.0/A(3) PE Fetch, division $B:=P[$X]*$S PE Fetch overlapped, Multiplication 63 9 ^r*i $C1:=$C1+1 $A:=RTR($C1,,A(3)) $S:=RTR(l,,$S) $X;=RTR(l,,$X) $R:=RTR(1,,$B) $A:=$A-$S $S:=1.0/$A %E $A:=P[$X]-$R $B;=$A*$S %Q, CU Addition overlapped PE Fetch, overlapped, Route Route Route Route Addition Division PE Fetch overlapped, Addition Multiplication 6 3 3 3 7 56 7 9 9Mn-i $A:=RTR($C1,,A(3)) $R:+$B $A:=$A*$B $C2:=$D6 $S;=P[$X](2)-$A P[$X](2):=$R PE Fetch overlapped, Route PE Register transfer overlapped Multiplication CU Register transfer overlapped PE Fetch overlapped, Addition PE Store overlapped 22 &1 $C2 : =$D2+$D3 ; $C1 : =$C1-1 $B $S $X =RTL(l,,$R) =RTL(1,,$S) =RTL(1,,$X) %E $A;=RTR($C1,,A(3)) $R : =$B #Q $A:=$A*$S $A;=P[$X](2)-$A $A;=$A=$R %T P[$X](2):=$S $B:=$S %Q $S:=$A %E 'CU Addition overlapped Route Route Route PE Fetch, partially overlapped Route PE Register transfer overlapped Multiplication PE Fetch overlapped, Addition Addition PE Store overlapped PE Register transfer PE Register transfer 3 3 3 10 9 7 7 1 1 U6(M-i) *C2:=$D1;$C3:=$D0 P(2):=RTL($C3,,P(2)) CU Register transfer overlapped PE Fetch, Route, PE Store 20 20 (m-dH SUBTOTAL i^^l-^farl +ko (M-l)f- Table 12. CRED* *The logic for the code to keep current values in the ADB's defined in the initial conditions is supplied in GLYPNIR, Appendix B. 60 the boundary conditions by -^ . Then apply (1+8.1), (1+8.2), the complex fast Fourier transform and (1+9) before. and after CRED. This algorithm, MFACR, is a modified FACR which solves (39) for Dirichlet's boundary conditions without using odd/even reduction. 3.5 Implementation of MFACR We will discuss MFACR in explaining how it can be programmed in ASK and how much time such a routine would take. Appendix F contains GLYPNIR code of the algorithm. Appendix A contains information on timing methodology and explains the notation used in Tables 9 through 12. Both ¥ and U are M+2 by N+2 mesh. ¥ and U are stored in straight storage in C and P, respectively. MFACR consists of five steps: 1. Set up storage area P 2. Fourier analysis uses f/ to calculate f/ 3. CRED uses y S to calculate U S h. Fourier synthesis uses U to calculate U 5. Restore boundary conditions and C. Step one places a linear combination of the boundary conditions and f/ in the interior of the storage area P. In the process the first row of U along with the second and N+lst row of ¥ are lost. Thus they are saved in temporary storage. This section of the algorithm takes (23M+102) T^irl + !3M clocks in ASK. Section 1 of Appendix F contains the GLYPNIR code. Fourier synthesis and analysis are performed on up to 6k columns simultaneously described above. They are presented in three parts. First equations (1+8.1) and (1+8.2) prepare the data for the complex 61 * fast Fourier transform. Table 9 contains suggestions on how these equa- as tions could be programmed in ASK and -supplies (57M-155) (FT clocks a time estimate for this step. Table 10 contains suggestions on how the complex fast Fourier transform could be programmed in ASK. A time esti- mate of \tt ^ Mfl ) Log (M+l) + 5 M-79) Ig^l + 3^M - 5^ is given for this step. Finally the implementation of equation (1+9) in ASK is discussed in Table 11 where a time estimate of 29 (M+l) tt- + 5M + 5 is presented. 53 The total time estimate for Cooley's algorithm is {-^-r- (M+l) Log (M+l ) + 91M - 205} frr-l + 39M - 1+9 clocks. The GLYPNIR code for Cooley's method is found in section 2 of Appendix F. MFACR uses the modified Thomas' method, equations (1+1+) , (1+5), and (1+7) to perform CRED on up to 6k rows simultaneously. Table 12 suggests how this method could be programmed in ASK and estimates how long that code would take, Uo(M-l) W- + (ll+0N-l+6) gr-| clocks. Section 3 of Appendix F contains the GLYPNIR code for CRED. The final step of MFACR consists of restoring the boundary condi- tions and the portions *+/ which have been altered. Section k of Appendix F contains the GLYPNIR code for this step. If programmed in ASK this step would take approximately 1+2 \zr\ clocks. The complete MFACR requires (53/2(M-l) Log (M+l) + 245M-306 } \jt-\ + (ll+0N-l+6) YqA + 91M - 98 clocks. The timing algorithm is taken from Appendix A. The loop numbers in Tables 9 to 12 correspond to the loop number in the GLYPNIR code in Appendix F. They are supplied to enable the reader to compare loops in the GLYPNIR code with the corresponding loops in the ASK code. 62 3.6 Implementation of FACR FACR, one level of odd/ even reduction would require (53/2(M-l) log (M+l) + U05M - k66) kf^j + (80M + 10U) f^fl + (TON - 23) Qf| + 8lM - Qk clocks. This figure assumes the odd columns are solved by Thomas' method with the e's calculated and stored in preconditioning. Thus the odd columns would take (8UM - k6) fTpo" clocks. Setting up the even columns would take 79M k^To" • Preparing the odd rows for solu- tion would take 55M \Tpdi clocks. CRED would take (70N-23) \-rrl clocks. Fourier analysis and synthesis would take (53/2(M-l) log (M+l) + 192M - 420) Ji2R + ^^M " ^ The extra storage manipulation rNi would take 1JM ytt] clocks. The rest of the algorithm is identical to MFACR. 3. 7 Summary of the Results on Direct Methods In programming an algorithm on ILLIAC IV, an attempt is made to adapt the restrictions of the algorithm to maximize storage and machine efficiency. For efficient storage on ILLIAC IV, rows stored across PEM should contain a multiple of 6k words. To maximize machine efficiency the number of values being computed simultaneously equals 6h. Hockney's method performs a Fourier analysis and synthesis on each column and performs CRED on each row. Thus to maximize machine efficiency on ILLIAC IV the number of interior rows and the number of interior columns must be multiples of 6k. Fourier analysis and synthesis using Cooley's method requires 2 - 1 points per column for Dirichlet's and Neumann's boundary conditions. If I > 6 then storing the columns across PEM would waste 63 words per column. Thus we will store each column in one PE. Now we have columns restricted to 2 +1 (2 ) for 63 Dirichlet's or Neumann's (periodic) boundary conditions. Rows are restricted to multiples of 6h to maximize storage efficiency. For periodic "boundary conditions cyclic reduction saves time and temporary storage over Thomas' method [Hockney, Page 150], and restricts the mesh to 2 columns. This restriction on mesh size still allows maxi- mum storage and machine efficiency. Cyclic reduction restricts the mesh to 2 +1 for Dirichlet's and Neumann's boundary conditions This restriction reduces storage efficiency for both boundary conditions and it reduces machine efficiency for Neumann's boundary conditions. Thomas' method and the modified Thomas' method have no restrictions on the number of columns. Thomas' method is faster than the modified Thomas' method but requires temporary storage. Refer to Table 8 for a more detailed comparison of the methods available for use by CRED. Hockney's FACR program handles the 9 different combinations of boundary conditions and performs one level of odd/even reduction in all cases. To maximize efficiency of such a program on ILLIAC IV different algorithms would be required for different mesh sizes and combinations of boundary conditions. As stated earlier, odd/even reduction saves computer time on ILLIAC IV for certain mesh sizes but not for others. Thus FACR on ILLIAC IV should have two algorithms, one which uses odd/even reduction and one that does not. Furthermore, the choice of a method to solve CRED depends on the storage requirements of the individual problem. It is the author's opinion that modified Thomas' Method is the best compromise between storage and speed for Neumann's and Dirichlet's boundary conditions, and cyclic reduction is the best for periodic boundary conditions. 6U If Fourier analysis and synthesis is applied to columns which have Neumann's boundary conditions then there must he 2 +1 interior rows to be evaluated by CRED. This means CRED must be executed < ■■;--<- J .-1 „I iul I 6k I (2 1 +vl 2 1 times. Note | „ | _ 7T +1 if I > 6 and thus for one execution of CRED, ILLIAC IV is working at l/6h of its capacity. The user can avoid this machine inefficiency by setting up the program so that Fourier analysis and synthesis is performed in the direction which has non-Neumann boundary conditions. On a M by N mesh the Fourier part of FACR requires the order of NMlogpM clocks when it is applied to the N columns in the mesh. CRED requires the order of NM clocks. By letting M be the smaller of the dimensions the user saves computer time. 65 k. USE OF ILLIAC DISK FOR NON-CORE CONTAINED MESHES When the problem "being run becomes too large to contain in core, the I/O time becomes an important consideration. First the I/O system for ILLIAC IV will be discussed and then its efficiency for a specific problem will be examined. k.l ILLIAC IV I/O System The ILLIAC IV Disk is a 15,600 K word memory. This memory is divided into 52 bands. Each band is divided into 300 pages, each con- taining l6 lines of PEM (l K words). Memory transfers between the ILLIAC IV Disk and PEM are restricted to pages. A data request can read or write up to 128 consecutive pages on one band. If the data is not consecutive or if it is spread over more than one band a separate request is needed for each string of consecutive pages. The time required for the disk to prepare to perform a data request is equivalent to the time to transfer two pages of data between the Disk and PEM. Thus if a data request reads or writes on page i and band j, the next data request should skip at least two pages and start at page i + 3 of band k, where j need not equal k. If the second data request wanted page i + 1, or i + 2, the request would have to wait for a revolu- tion of the disk before it could be executed. The transfer rate between PEM and Disk is 133 usee per page and the disk revolves once every kO msec. One revolution of the disk is equivalent to 61+0,000 ILLIAC IV clocks ^.2 1/0 for the Bernard-Rayleigh convection problem Now let us examine the Bernard-Rayleigh convection problem where the following equations are used to solve for temperature, T, pressure, P, and velocity components u and w along with the X and Z axes: 66 £ + u » + „ f = i- V 2 ! (50) 9t 9x 9z P r 3n 3n L 3n _ R a 9T 2 , » r-+ ur- + w — = — TT V n . (51) 9t 9x 9z P 9x r n = |* _ |H = V 2 * (52) 9X dZ where t is time, R is the Rayleigh number, P is the Prandtl number, and ^ is the stream function defined by 9i1j 9z ' 9x The finite difference schemes employed to solve these equations require five meshes: T (t) , T {x ' l \ r/ , n^ and * (x) . Each time step of the convection process has three main parts; (x+l) (t+1) (t+1) calculate n , calculate T and calculate ¥ „<^> is calculated using, W ., ^ jff) , jg^ .^W^. T ^ is calculated using T^., T^, T^T 1 ' *j$ >Jtl , l&J^ «^»i^l- I Using all of r\ J ,¥ is calculated. The process will be divided into two parts. First the prognostic ain ri (t+1) equations (50 ) and (5l) are solved to obtain ri and T . Then Pois son's equation, (52), is solved for \F The prognostic equations require five meshes while Poisson's equation requires two meshes if solved by an iterative method and one mesh if solved by a direct method. Thus non-core contained problems can be divided into two types. In the first type, the meshes required for solving Poisson's equation are core contained but some of the meshes required for solving the prognostic equations are not contained in core. 67 In the second case, the meshes are so large that none of the meshes can be core contained. In [Ogura, et al] the first type of problem was studied in which SOR was used to solve Poisson's equation. If a direct method had been used in that study, the maximum mesh size could be doubled. In this study the second type of I/O problem will be examined where MFACR is used to solve Poisson's equation. For an example we will use 512 by 512 meshes. P. . where 1 < i < 6U and < j £ 3 is a page of a mesh containing all points a where 8. - 8 < k < 8. and 128 j < £ < 128 j + ]28. If the prognostic equations are solved separately using ASK code they are found to be about 50% I/O bound. To decrease the I/O bound the Fourier analysis of MFACR will be performed along with the calculation of the prog- nostic equations. This will be followed by CRED, with the final step being Fourier synthesis. The data will be stored in blocks 22 pages long. Block number I, B contains P where < j < 3 of n , T ( , n ~ , T (x ~ , and }A T+1 /3J Row j of vjAt J 5) lg written 3 data blocks before Row I of ^ T+1 /3) during CRED.* The entire time step required 23 revolutions and 110 pages on the 2l+th revolution. This takes 932 milliseconds. The calculation could be per- formed in 333 milliseconds. Thus the process is 6k% I/O bound. A large portion of this I/O bound is created by the solution of Poisson's equation. The last two steps of MFACR required 13 revolutions of the disk while the calculations for those steps required the amount of time for 2^ revolut ions . * (t+1/3) y contains the harmonics calculated by Fourier analysis. (t+2/3) V contains the harmonics calculated by CRED. 69 To minimize total computer time a problem takes, a programmer must adapt algorithms to disk maps which give minimum I/O time. With I/O being such a dominant factor in the above problem the use of library- subroutines becomes minimal because programs need to be customized to minimize I/O. Figure 6. A Disk Map of 512 by 512 Meshes Page 1 2 3 I* 5 6 7 8 9 10 11 12 13 Ik 15 16 IT 18 19 20 Band Band 1 (t+1) 5,0 (t+1) 5,1 (t+1) 5,2 (t+1) 5,3 (t+1) 5,0 (t+1) 5,1 (t+1) 5,2 (t+1) Band 2 T T n 5,3 M "5,0 ,(t) "5,1 t) 5,2 C (T) "5,3 't) 5,0 l (T) 5,1 i (T) 5,2 i (t) 5,3 ¥ (t4) 5,0 J ,(x+i) 5,1 J (T+i) Y 3 5,2^ u;(T+T) 5,3 J Band 3 (t+1) n l8,0 (t+1) n l8,l (t+1) n l8 5 2 (t+1) n l8,3 ,(t) 18,0 (t) 18,1 (t) 18,2 (t) 18,3 n (T) n l8,0 (t) n i8,l n (T) n i8,2 \8,3 18,0 18,1 18,2 ,(-4) 18,3 Band 1+ Band 5 Band 6 T T. (t+i: 12,0 T n (t+1) 25,0 (t+1) 25,1 (t+1) 25,2 (t+1) 25,3 (t+1) 25,0 T T (t) 31,0 (t) 31,1 (t) 31,2 (t) 31,3 '31,0 (t) '31,1 (t) 31,2 >) 71 31, 3 31,0 31 ,(t+4) 31,2 y( T 4) 31,2 n 7 T ,(t+1) "38,0 ,(t+1) "38,1 (t+1) 38,2 t (t+D X 38,3 (t+1) n 38,0 (t+1) n 38,l (t+1) n 38,2 (t+1) n 38,3 t (t) 38,0 Band 7 n U+,0 (t) \k,l (t) n UU,2 n (T) J A) \kj V,3 Band T T T T ,(t+1) '51,0 ,(t+1) '51.1 (t+1) 51,2 (t+1) 51,3 (t+1) 51,0 (t+1) ^51,1 (t+1) ] 51,2 (t+1) l 51,3 ,(t) "51,0 (t) 51,1 (t) 51,2 (t) 51,3 1 (t) '51,0 Band 9 f (T*4) 57,0 M } T 57,l 57,2 1 1\ 57,3 Ban 10 pCxk ~6U= l 6h3 n 6^ \ki m(l T 61|D m( '( L 6^3 n 6li3 n 6U (1 n 6lj2 (t n 6L *. 61. t Ct+1 ) 12,1 m(T+l) 12,2 t (t+D 12,3 (t+1) 12,0 t+1) 12,1 T+l) 12,2 T+1) 12,3 x) 12,0 (t) 12,1 (t) 12,2 t) 12,3 n (T) 12,0 (t) n l2,l (t) n 12,2 (t) n 12,3 ¥ (t+ 3 } r 12,I vw 3 *12,3 T T, T (t+1) 19,0 (t+1) 19,1 (t+1) 19,2 (t+1) 19,3 (t+1) n 25,l (t+1) n 25,2 (t+1) n 25,3 m(T) 25,0 t (t) 25,1 t (t) 25,2 T (x) 25,3 n 25,0 n 25,l "25,2 n 25,3 1 7 i 25 25 (t+i) f 3 r 25,2" (t+t) i' 3 "25,3 ,(t+1) '32,0 ,(t+1) "32,1 ,(t+1) 32,2 ,(t+1) '32,3 (t+1) '32,0 (t+1) 1 32,1 (t+1) '32,2 (t+1) ^2,3 t (t) t (t) i 38,2 t (t) 38,3 n (T) n 38,0 n 38,l >) ^38,2 ^38,3 T 38,2 ? 51,1 51,2 51,3 (t+1) 58,0 (t+1) 58,1 (t+1) 58,2 (t+1) 58,3 (t+1) n 58,0 (t+1) n 58,l (t+1) n 58,2 (t+1) n 58,3 (t) 58,0 (t) 58,1 (t) 58,2 (t) 58,3 (t) n 58,0 (t) n 58,l (t) n 58,2 (t) n 58,3 m( T+ 3") 71 5. CONCLUSIONS Table 13 compares MFACR and FACR with SOR and ADI for Dirichlet's boundary conditions. The direct methods use modified Thomas' method to perform CRED and Cooley's methods to perform the Fourier part of the algorithm. Unless an excellent initial guess is supplied for the itera- tive methods, the direct methods are much faster. For maximum machine efficiency, MFACR requires 64k by 2 +1 meshes, FACR requires 128k by 2 +1 meshes, ADI requires 64k by 641 meshes, and SOR requires 64k by j meshes. As seen in section 2, SOR can partition its mesh to obtain maximum machine efficiency. If the mesh for one of the other methods did not meet the row restrictions, the mesh could not be partitioned to meet the restrictions. A possible solution would be to solve a number of meshes at the same time. (MESH SIZE M+2 by N+2 FACR MFACR SOR PER ITERATION ADI PER ITERATION 33 by 63 6.51 or 25300 5.71 or 22400 I or 3900 3.81 or 14900 65 by 63 5.31 or 49500 4 .91 or 39100 I or 7900 2.51 or 20100 65 by 127 3.71 or 59500 4.71 or 74000 I or 15900 2.61 40700 129 by 127 3. 81 or 122000 4.91 or 156000 I or 32000 2.71 or 87700 Table 13. A Comparison of Methods with Respect to Time* Two numbers are supplied. The number times I is the number of iterations which could be performed in an equivalent amount of time. The other number is the number of clocks required on ILLIAC IV. The times given for SOR in section 2.3 don't take into account the time required to check for convergence. To calculate the times for SOR in Table 13,20% has been added to the times in section 2.3 to account for convergence checking. 72 Direct methods require only one mesh in core. The mesh initially con- tains the interior source points and the boundary conditions. Throughout the algorithm, the mesh is used as temporary storage with the values in the mesh set equal to the solution in the final step. Iterative methods require "both the source and the solution mesh at every iteration. 73 REFERENCES Birkoff, G. and MacLane, S. A Survey of Modern Algebra . New York: The MacMillan Company, 1962. Cooley, J. W., et al. "The Fast Fourier Transform Algorithm: Program- ming Considerations in the Calculation of Sine, Cosine, and LaPlace Transform." Journal of Sound and Vibrations , vol. 12, no. 2, June 1970. Cu thill, E. H. and Varga, R. S. "A Method of Normalized Block Iteration." Journal Assoc. Comput. Mach . 6 (1959): 236-2UU. Denenberg, S. A. "An Introduction Description of the ILLIAC IV System. " Center for Advanced Computation Document no. 10. Urbana: University of Illinois, July 1971- Door, Fred W. "The Direct Solution of the Discrete Poisson Equation on a Rectangle." SIAM Review , vol. 12, no. 2 (1970): 2U8-263. Ericksen, James H. "A Survey of Iterative Methods for Solving Poisson' s Equation and Their Adaptibility to ILLIAC IV." Master's thesis. Urbana: University of Illinois, 1972. Forsythe, G. E. and Wason, W. R. Finite-Difference Methods for Partial Differential Equations . New York! John Wiley and Sons, Inc., i960. Hockney, R. w. "The Potential Calculation and Some Applications." Methods in Computational Physics , 9 (1970): 136-211. "ILLIAC IV Systems Characteristics and Programming Manual. " Burroughs Corporation Document no. 66000B, Paoli, Pennsylvania, 1969- Lawrie, D. H. "GLYPNIR Programming Manual." ILLIAC IV Document no. 232, Department of Computer Science. Urbana: University of Illinois, August 1970. Ogura, M., M. S. Sher, and J. H. Ericksen. "A Study of the Efficiency of ILLIAC IV in Hydrodynamic Calculations." Center for Advanced Computation Document no. 59- Urbana: University of Illinois, December 1972. Rudsinski, Lawrence E. "Tranquil Code for the MSOR Method for Solving LaPlace' s Difference Equation." ILLIAC IV Document no. 208. Urbana: University of Illinois, December 1968. Stevens, James E., Jr. "A Fast Fourier Transform 'Subroutine for ILLIAC IV." Center for Advanced Computation Document no. 17. Urbana: University of Illinois, July 1970. Tod, J. Survey of Numerical Analysis . New York: 1962. Wachspress, E. L. Iterative Solution of Elliptic Systems and Applications to the Neutron Diffusion Equations of Reactor Physics . Englewood Cliffs, New Jersey: I966. Ik Widlund, Olof B. "On the Use of Fast Methods for Separable Fintie Difference Equations for the Solution of General Elliptic Problems.' New York: Courant Institute of Mathematical Sciences. Young, D. M. Iterative Solution of Large Linear Systems . New York: Academic Press, 1971- A-l APPENDIX A TIMING METHODOLOGY The procedure for estimating the amount of time required by an algorithm is explained below. Care has "been taken to insure that the method of calculating the computer time does not give one algorithm an unwarranted advantage over another. In determining the amount of time an algorithm takes on a computer we must consider the language in which the algorithm is programmed. In comparing GLYPNIR to ASK, one finds that the average GLYPNIR code takes about twice as long as the ASK code for the same problem. Two major reasons for this difference are memory fetching and indexing. The code generated by GLYPNIR for indexing can be improved considerably by rewriting it in ASK. GLYPNIR does fetching and storing of data between PEM and PE registers that can be eliminated by efficient use of the PE registers in ASK. GLYPNIR code is much easier to write than ASK. One must choose between simplicity and speed. However, GLYPNIR permits the programmer to write sections of code in ASK. Thus the programmer can write code in GLYPNIR and then rewrite the statements which are executed most often in ASK. In estimating the time for the different methods, we have timed the statements which are executed in the inner loops of the iterative method. Since these statements take most of the computer time they should be written in ASK. We use the GLYPNIR statements as an outline of the method but the time estimates reflect efficient ASK code , not the code that GLYPNIR would generate. To show the reader how we arrived at our time estimates we have included a table with assignment statements, operations which are not overlapped and the clocks required for each method, A-.: The assignment statements are written in a combination of GLYPNIR and ASK. The parts of the statements -which are not used in GLYPNIR are defined in Table lU. A summary of the assumptions used in obtaining the time esti- mates is found in Table 15. NOTATION MEANING NOTATION MEANING $A PE Register A $Ci ACAR i for i=0,l,2,3 $B PE Register B U(i) U indexed by ACAR i $R PE Register R u[$x] U indexed by register X $S PE Register S D[$X](i) both $X and $Ci indexing of U $X PE Register X $Di ADB storage location i for i=0, . -,6 3 Table 1^' Assignment Statement Notation Every iterative algorithm we consider in this study will have two parts: Phase 1 will be used to improve 6h points simultaneously; Phase 2 will improve up to 63 points simultaneously. The machine efficiency of Phase 1 is greater than the machine efficiency of Phase 2. If one used only Phase 1 on certain meshes the extra time needed to bring the data to the PE's would exceed the time saved by improving machine efficiency. Thus the fastest cede is obtained by a combination of Phase 1 and Phase 2. 1) CU instructions are overlapped completely by a combination of FINST/ PE instructions and PEM fetches. 2) Memory fetching is overlapped as much as possible. 3) Seven PE clocks are used to load (Jst-drare) a PE register from (to) PEM. k) One PE clock issued to load a PE register from the CU. 5) Transfer between PE registers require one clock. 6) Each GRABONE will be considered to be an ASK LOAD instruction ( 10 clocks). Table 15. Assumptions in Timing The author feels these timing assumptions should give times within 20% of the actual times the algorithms would take on ILLIAC IV. A- 3 To aid in comparing times of different methods we will define an execution of Phase 1 to he the improvement of 6k points simultaneously hy Phase 1. Similarly, an execution of Phase 2 is the improvement of k points where Phase 2 improves k points simultaneously. B-l APPENDIX B MAJOR STEPS OF FACR AND MFACR FOR DIRICHLET'S BOUNDARY CONDITIONS ' Given the meshes ¥ and U with mesh points at locations ¥. and U where < i ^ M and ^ j ^ N the five point difference scheme approxima- tions of Poisson's equation becomes Vl.t - 2 "s.t + Vl.t , °B.t-l - 2U S,t + "s.t+1 T ,■ ' h 2 h 2 »•* ' x y To solve (B.l) FACR works as follows. Odd/even reduction (see Section 3.1 ) calculates y for the even columns of ¥. Equation (Ul ) from Section 3.1 can be expanded into the following eauation for individual mesh points. 1 - (U i,t-2 + U i,t + 2 } -:V U i- 2 ,t + V2,t } y * *>& + *) «v l!t + w - f" ^ + r* + M u ^ = f "i.* (B ' 2) h h X x h h h x x y where * . = ¥ - -2— ft + V 1 + ?l -X_ * il vu 4. u/ (B.2.1) *•* i.t-1 h 2 ( Vl,t Vl,t } 2 K2 + 4 *i,t + *l,t-l Fourier analysis calculates the Fourier harmonics Y S and ^* defined by M-l ¥ i,t m^ * k ,t SIW ir • ( B -3) Combining (B.2) and (B.3) we obtain A, t ■ I T [s ™ *f [fs \,t. a ♦ w - j4 t + w y x B-2 5 The Fourier harmonics for U, U are defined as follows: M-l K. — X and thus M-l irilk TT s Combining (B.5), (B.6) and (B.k) we obtain iS 1 /TT s , TT s h 2 = _±_ (u s + u s ) - 6 -V + -~ + -—. | it i,t h 2 l i,t+2 i,t-2 ; ^T h 2 h 2 J i,L y * x x y r M-l_ . . r h 2 M-l ,_ _» , oU M k=lL M 1 ii 1=1 £ ' £ ' rh2 1 h h / 1= x y j (SIN ^%^%: + + SIN ^-^ U* j)l (B.7) Using the facts that Sin(a±b) = Sin a Cos b ± Sin b Cos a and M-l r _ . 7r£k _ . TTJk N - . . -, o iv/r n -u 2 Sin — Sin — = g 6 i £ for £>1 = lj 2 > •"» M_1 where k=l ' f if i f i, 1^1 if i = J system £ 5 i ^ 1 if i = p ve can de couple (B.T) into a tridiagonal linear h 2 /h f i,t.= ^^, t+ 2 +u i,t- 2 )- K^sf .b^^JcobS y ' x \ x x h 2 Ii h d h 2 I x x y CRED solves (B.8) for U S . B-3 g Fourier synthesis calculates U from U using (B.6). This gives us the values for U on the even columns. Using the values of U on the even columns we can solve (B.l) for the values of U on the odd columns using a method which solves tridiagonal linear systems. In summary FACR performs the following algorithm: 1. Given ¥ compute ¥ for the even columns using (B.2.1) and overwrite y with y on the even columns. 2. Using 4* compute ¥ according to (B.3) and overwrite y 1 with Y . O C C C! 3. Using y solve (B.8) for IT and overwrite ^ with U . h. Using u compute U according to (B.6) on the even columns and over- write U with U on the even columns. 5. Using the values of U on the even columns solve (B.l) for the values of U on the odd columns and overwrite V with U on the odd columns. MFACR is similar to FACR but the odd/even reduction step is omitted. MFACR begins with Fourier analysis which calculates the Fourier harmonics y for f using o M-l y ''i,t "" M , *, \,t "*"■" M k=l From B. 9 v e obtain M-l /U - 2U + U i,t M v £ n M V h 2 SIN lki (B. 9 ) k=l h x U s,t-1 - 2U s ,t +U s,t + l \ (B.10) h 2 y B-U Using the same methods employed to obtain (B.8) from (B.T)we obtain (B.H) from (B.10). s 1 / s s v , / 2 „_ TTi 2 2 \ s / *i,t " ^2 (u i,t + i + "i.t-i' + ( ^T cos -m " — " -zH,t f 8 - 11 ' y y y x/ CRED solves (B.ll) for if . i »t Finally (B.l) is used to calculate U for the entire mesh. 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O • O o O >- Or - UJ U CO z TOO D < O z x r •-< x i— • o z o or >-> o < or o o Q. o ^ II r- Z o i- o ^ o o u UJ .-. »— li or UJ UJ UJ O IM II t— i o to en o- or < o i o a: «r o: tr •• uj o UJ O _l u. a: -i o a u_ a uj uj UJ o o dl a a. T O 3 r> o o z CO CO Q CO o or o • » o z 00 ■-> o co •* CO o> uj o o O O or ♦ o. UJ Q CO O O CJ Z O CO o O UJ or co o 3 o (\J » • o o CO O LJ CO or il a UJ1< < I- _J 1 Z UJ "•* UJ nun 0: ujq •-• v o UJ _J UJ CO o or v — o >- i -< 5" UJ 3" • • UJ »-i >-HK or o < o z or o 00 Q >- >— UJ (- _» t- < < _J X *-• a — a. < CO o 1 UJ CJ UNCLASSIFIED Security Claaaiflcatiort DOCUMENT CONTROL DATA -R&D (Security etaaatltoatlanaljltla, Jfcaa> at mMm€tmml JMSlartsj — wtoWw mmut ma wl«W ■*<■> tfw> »wr«l) report la ctaaaltlatt) originating activity (Carporata author) Center for Advanced Computation University of Illinois at Urbana-Champaign Urbana, Illinois 61801 am. REPORT SECURITY CLASSIFICATION UNCLASSIFIED 2ft. GROUP REPORT TITLE Iterative and Direct Methods for Solving Poisson's Equation and Their Adaptability to ILLIAC IV descriptive MOTES (Typa at rtmttt ar\4 tnalualra motaa) Research Report autmorisi (Pint nam : mtmaVa Initial, laat nm m m) James H. Ericksen REPORT OATE December 20, 1972 7*. TOTAL NO. OP PACE* life 76. NO. OF REFS 12 «. CONTRACT OR 6RANT NO. DAHC04 72-C-OOOl b. PROJECT NO. ARPA Order No. 1899 •a, ORIGINATOR'S REPORT NUMSIRIII CAC Document No. 60 M. OTHER REPORT NOI1I (Any othar numbarm thml may ba ammianad thla rapart) 0. DISTRIBUTION STATEMENT Copies may be requested from the address given in (l) above I. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U. S. Army Research Office - Durham Duke Station, Durham, North Carolina )D ,?~„1473 UNOT.AHSTFTC Security Claaaifii cation UNCLASSIFIED Security Classification KEY WOROI Glypnir Poisson equation solvers ASK Numerical integration Ordering and partial differential equations Computer efficiency and benchmarks Input/ Output UNCLASSIFIED Security Classification IOGRAPHIC DATA ET 1. Report No. UIUCDCS-R-72-57U tie and Subtitle Iterative and Direct Methods for Solving Poisson's Equation and Their Adaptability to ILLIAC IV ithor(s) James H. Ericksen 3. Recipient's Accession No. 5. Report Date December 20, 1972 6. 8. Performing Organization Rept. No. rrforming Organization Name and Address Center for Advanced Computation University of Illinois at Urb ana-Champaign Urbana, Illinois 6l801 10. Project/Task/Work Unit No. 11. Contract /Grant No. DAHCO^ 7 2 -C -0001 ponsoring Organization Name and Address U. S. Army Research Office - Durham Duke Station Durham, North Carolina 13. Type of Report & Period Covered Research Report 14. upplementary Notes bstracts This paper examines iterative and direct methods for solving Poisson's equation with regard to their adaptation to ILLIAC IV. S0R, SL0R, ADI, and FACR are programmed in GLYPNIR. Detailed suggestions on ASK code for these methods are also supplied. FACR, Fourier Analysis and Cyclic Reduction, is the fastest method on rectangular meshes. S0R, Successive Over Relaxation, seems to-be the most promising for nonrectangular meshes. The methods are between thirty and forty- five times faster on ILLIAC IV than on a serial machine with speed equal to one of the ILLIAC IV PEs (Processing Elements). ey Words and Document Analysis. 17o. Descriptors Glypnir Poisson equation solvers ASK Numerical integration Ordering and partial differential equations Computer efficiency and benchmarks Input/Output Identifiers/Open-Endcd Terms "OSATI Field/Group ailability Statement Distribution unlimited 19. Security Class (This Report) UNCLASSIFIED 20. Security Class (This Page UNCLASSIFIED 21. No. of Pages 109 22. Price 1 NTIS-3B ( 10-701 USCOMM-DC 40329-P7 1 OUNO^