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strides. In general, ERL has the useful property that the size <strong>of</strong> the set being representedcan be easily computed. To find an algorithm <strong>for</strong> finding the size <strong>of</strong> such a group E, I firstprove a theorem.Theorem 3. Let there be two d-dimensional RCSLMADs L 1 and L 2 defined over the samedomain D with equal strides p k in each dimension k. If the bases b 1 and b 2 <strong>of</strong> the twoRCSLMADs are such that 0 ≤ b 2 − b 1 ong>of</strong>. I prove the theorem by contradiction. Let there be two d-dimensional tuples V 1 ɛDand V 2 ɛD such that L 1 (V 1 ) = L 2 (V 2 ). For the sake <strong>of</strong> contradiction, I assume that V 1 ≠ V 2 .Let m be the smallest dimension in which V 1 and V 2 differ. The rest <strong>of</strong> the pro<strong>of</strong> is similar tothe pro<strong>of</strong> given in theorem 1. First, consider the case where V 1 (m) < V 2 (m). We substitutethe values <strong>of</strong> V 1 and V 2 in the definitions <strong>of</strong> L 1 and L 2 .Then we apply the conditionp m > p 1 − 1 + ∑ dk=m+1 p k ∗ (u k − 1). Following the logic <strong>of</strong> pro<strong>of</strong> <strong>of</strong> theorem 1, using thecondition it can be proven that L 1 (V 1 ) < L 2 (V 2 ). There<strong>for</strong>e, we arrive at a contradictionand the theorem is proved <strong>for</strong> this case. Second, consider the case V 1 (m) > V 2 (m). In thiscase it can be proven that L 1 (V 1 ) > L 2 (V 2 ). Again we arrive at a contradiction and thetheorem is proved.Given the previous theorem, computing the number <strong>of</strong> distinct elements in an d-dimensionalERL E composed <strong>of</strong> n RCSLMADs is straight<strong>for</strong>ward.Theorem 4. Let E be a d-dimensional ERL composed <strong>of</strong> n RCSLMADs L 1 , L 2 , .., L n withbases b 1 , b 2 , ..., b n respectively. Then the number <strong>of</strong> distinct elements <strong>of</strong> E is given by q ∗d∏u k where q is the number <strong>of</strong> distinct elements in the list {b 1 , b 2 , b 3 , .., b n }.k=1Pro<strong>of</strong>. The theorem is a corollary <strong>of</strong> the previous theorem.5.3.2 One-dimensional RCSLMADs with common strideConsider two one-dimensional RCSLMADs L 1 , L 2 having the same stride m and definedover the domain D.L 1 = b 1 + m ∗ i (5.33)L 2 = b 2 + m ∗ i (5.34)0 ≤ i ong>of</strong> generality assume that b 1 ≤ b 2 . As was shown earlier, RCSLMADs canbe visualized as simple-strided accesses into an imaginary C array. If there is more than oneRCSLMAD, we can attempt to place both <strong>of</strong> them in the same imaginary C array. In this42
case, assume we had a C array starting at some arbitrary positive integer b 0 ≤ b 1 . 1RCSLMADs can be rewritten as:TheL 1 = b 0 + m ∗ (i + t 1 ) + r 1 (5.36)t 1 = ⌊(b 1 − b 0 )/m⌋ (5.37)r 1 = (b 1 − b 0 )%m (5.38)L 2 = b 0 + m ∗ (i + t 2 ) + r 2 (5.39)t 2 = ⌊(b 2 − b 0 )/m⌋ (5.40)r 2 = (b 2 − b 0 )%m (5.41)0 ≤ i < u (5.42)Further rewriting:L 1 = b 0 + m ∗ j + r 1 (5.43)t 1 ≤ j onstruct an RCSLMAD L 3 that is a superset <strong>of</strong> L 1 by increasing the domain <strong>of</strong> L 1 .L 3 = (b 0 + r 1 ) + m ∗ j (5.47)0 ≤ j onstruct an RCSLMAD L 4 as a superset <strong>of</strong> L 2 .L 4 = (b 0 + r 2 ) + m ∗ j (5.49)0 ≤ j ong>of</strong> L 3 and L 4 is equal to |r 1 − r 2 | and |r 1 − r 2 | < m because r 1 < m andr 2 < m. The union <strong>of</strong> L 3 and L 4 is there<strong>for</strong>e a one-dimensional ERL E defined over domainD ′ with stride m and with bases b 0 + r 1 and b 0 + r 2 . Since we have not yet chosen b 0 , weshould attempt to choose a b 0 such that the set D ′ is the smallest. The integer b 0 = b 1 gives1 The new base b 0 will later be chosen to minimize the domain <strong>of</strong> an RCSLMAD that combines thememory references <strong>of</strong> L 1 and L 2 .43
Page 3 and 4: Examining CommitteeJose Nelson Amar
Page 5 and 6: AbstractModern Graphics Processing
Page 8 and 9: 5.3.3 Multidimensional RCSLMADs wit
Page 10 and 11: List of Figures2.1
Page 12 and 13: RAMRISCSIMDSPSPMDSSETEXTPUVDVLIWbDE
Page 14 and 15: much lower than equivalent C progra
Page 16 and 17: Chapter 2AMD RV770 architecture <st
Page 18 and 19: Figure 2.1: RV770 Block Diagram. So
Page 20 and 21: 2.2 AMD CAL Programming modelVariou
Page 22 and 23: a context. The predefined names are
Page 24 and 25: 2.3 Hardware execution model2.3.1 P
Page 26 and 27: on the texture unit to bring the da
Page 28 and 29: float4 temp1 = i1[i][threadId.y/4];
Page 30: Chapter 3Python an
Page 33 and 34: Unlike C, NumPy does not impose any
Page 35 and 36: Windows. The module initialization
Page 37 and 38: 3.5.1 Type annotationsType declarat
Page 39 and 40: UnPython generates
Page 41 and 42: Python source code
Page 43 and 44: and optimizations detailed later in
Page 45 and 46: Chapter 5Array access analysisArray
Page 47 and 48: LMADs can represent a wide variety
Page 49 and 50: Figure 5.2: Visualization o
Page 51 and 52: Algorithm 1 solves the data transfe
Page 53: 0 ≤ j < 5 (5.26)In the example, t
Page 57 and 58: solve for unknown
Page 59 and 60: approach, the number of</st
Page 61 and 62: Chapter 6Loop transfor</str
Page 63 and 64: Algorithm 6 perfor
Page 65 and 66: with suitable GPU code annotations.
Page 67 and 68: Table 7.3: Execution time f
Page 69 and 70: time and the execution time on the
Page 71 and 72: suite. The serial version o
Page 73 and 74: However Shedskin does not support e
Page 75 and 76: thread. Device functions are writte
Page 77 and 78: Chapter 9ConclusionsThis thesis int
Page 79: [14] Francois Labonte, Peter Mattso

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