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[(Lemma)-302(3.)]TJ
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ET
q
1 0 0 1 184.072113 313.6994934 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_2 1 Tf
9.9626 0 0 9.9626 184.0721 304.5742 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
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[(R)-22(O)-61(O)-62(F)]TJ
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[(:)-461(It)-325(suf)25(\002ces)-326(to)-325(consider)-325(the)-326(case)-325(where)]TJ
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[(ul)1(p\()]TJ
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[(ul)1(p\()]TJ
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(\000)Tj
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[([u)1(lp\()]TJ
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[(\))-222(+)-222(ul)1(p\()]TJ
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(p)Tj
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[([u)1(lp\()]TJ
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(a)Tj
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[(\))-222(+)-222(ul)1(p\()]TJ
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[(ul)1(p\()]TJ
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[(\))-278(=)-277(ulp)1(\()]TJ
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[(we)-250(get)]TJ
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(p)Tj
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[(ul)1(p\()]TJ
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[(b)-278(<)]TJ
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(p)Tj
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(+1)Tj
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[(ul)1(p\()]TJ
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[(ul)1(p\()]TJ
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[(\))-222(+)-222(ul)1(p\()]TJ
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[(\(Lemma)-250(2\))-250(and)-250(the)-250(lemma)-250(holds.)]TJ
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[(Otherwise)-245(we)-245(can)-246(assume)-245(without)-245(loss)-245(of)-246(general)1(ity)-246(that)]TJ
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9.9626 0 0 9.9626 103.5039 300.674 Tm
[(\(1\))-222(+)]TJ
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Q
BT
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6.9738 0 0 6.9738 225.118 292.4398 Tm
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6.9738 0 0 6.9738 239.9703 292.4398 Tm
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(x)Tj
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6.9738 0 0 6.9738 252.8555 292.4398 Tm
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[(From)-193(Equation)-194(\(1\))-193(we)-194(can)-193(see)-193(that)]TJ
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6.9738 0 0 6.9738 203.6932 257.0611 Tm
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9.9626 0 0 9.9626 209.1162 258.5555 Tm
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(\006)Tj
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[(=)-278(1)]TJ
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(x)Tj
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6.9738 0 0 6.9738 78.4173 245.1059 Tm
(j)Tj
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9.9626 0 0 9.9626 82.6151 246.6003 Tm
[(\))-278(=)]TJ
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9.9626 0 0 9.9626 142.4603 216.7898 Tm
(=)Tj
4.257 0.677 Td
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ET
q
1 0 0 1 154.171814 219.0812988 cm
0 G
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S
Q
BT
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(\000)Tj
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(x)Tj
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(2)Tj
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0 -0.89 TD
(i)Tj
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(\))Tj
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(P)Tj
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(0)Tj
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(\()Tj
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(x)Tj
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(i)Tj
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6.187 0 Td
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[(Since)-428(we)-428(can)-429(compute)]TJ
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(i)Tj
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9.9626 0 0 9.9626 170.9848 186.6245 Tm
[(to)-428(arbitrary)-428(precision,)-473(we)-428(can)]TJ
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[(use)-305(Equation)-304(\(2\))-305(to)-304(compute)]TJ
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11.855 0 Td
(w)Tj
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6.9738 0 0 6.9738 183.9881 173.1749 Tm
(i)Tj
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9.9626 0 0 9.9626 190.3391 174.6693 Tm
[(with)-305(arbitrary)-304(accurac)15(y)-305(as)]TJ
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[(well.)-608(Recall)-349(that)]TJ
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9.9626 0 0 9.9626 147.1044 162.7141 Tm
[(is)-349(kno)25(wn)-349(e)15(xactly)-350(and)-349(we)-349(can)-349(get)-349(an)]TJ
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[(error)-303(bound)-303(on)]TJ
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[(\(kno)25(wn)-303(as)]TJ
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4.469 0 Td
[(running)-303(err)45(or)]TJ
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[(\))-303(using)-303(al-)]TJ
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[(gorithm)-237(5.1)-236(from)-237([5,)-239(p.)-306(95].)-305(In)-237(the)-237(rest)-236(of)-237(this)-236(paper)-237(we)-237(will)]TJ
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[(is)-284(computed)-283(as)-284(the)-283(rounded)-284(to)-284(nearest)]TJ
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[(of)-250(the)-250(e)15(xact)-250(v)25(alue:)]TJ
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(j)Tj
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(\000)Tj
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(w)Tj
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(i)Tj
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9.9626 0 0 9.9626 165.6854 87.8339 Tm
[(j)-278(\024)]TJ
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1.731 0.677 Td
(1)Tj
ET
q
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0 G
0.398 w 10 M 0 j 0 J []0 d
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S
Q
BT
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9.9626 0 0 9.9626 182.9317 81 Tm
(2)Tj
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[(ul)1(p\()]TJ
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(c)Tj
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(i)Tj
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(\))Tj
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(:)Tj
ET
q
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0.797 w 10 M 0 j 0 J []0 d
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236.25 0.398 l
S
Q
BT
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9.9626 0 0 9.9626 317.5019 710.3462 Tm
[(Algorithm)-250(2)]TJ
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5.444 0 Td
[(Gauss-Le)15(gendre)-250(inte)15(gration)]TJ
ET
q
1 0 0 1 317.5019073 705.7984009 cm
0 G
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S
Q
BT
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(I)Tj
7.9701 0 0 7.9701 338.5031 693.2805 Tm
[(N)-62(P)-61(U)-62(T)]TJ
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(:)Tj
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(\))Tj
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[(;)-166(f)-52(;)]TJ
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(b)Tj
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(\))Tj
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[(;)-166(n)-1828(.)]TJ
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4.011 0 Td
(where)Tj
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(i)Tj
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9.9626 0 0 9.9626 526.9229 693.2805 Tm
[(are)-250(the)]TJ
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[(weights)-250(and)]TJ
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(i)Tj
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[(is)-250(de\002ned)-250(in)-250(\2472.4.)]TJ
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(O)Tj
7.9701 0 0 7.9701 342.3786 667.7667 Tm
[(U)-62(T)-61(P)-62(U)-61(T)]TJ
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(:)Tj
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(b)Tj
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-0.092 -0.253 Td
(I)Tj
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0.518 0 Td
[(,)-322(a)]TJ
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1.323 0 Td
(p)Tj
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[(-bit)-307(approximation)-308(of)]TJ
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9.48 0.806 Td
(R)Tj
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6.9738 0 0 6.9738 496.6934 673.3292 Tm
(b)Tj
-0.278 -1.306 Td
(a)Tj
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(f)Tj
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0.597 0 Td
(\()Tj
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(x)Tj
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0.96 0 Td
(\)d)Tj
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0.944 0 Td
(x)Tj
/T1_0 1 Tf
0.879 0 Td
(with)Tj
-20.236 -1.2 Td
[(error)-250(bounded)-250(by)-250(Theorem)-250(3.)]TJ
7.9701 0 0 7.9701 323.2565 643.8564 Tm
(1:)Tj
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9.9626 0 0 9.9626 334.4385 643.8564 Tm
[(f)25(or)]TJ
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1.502 0 Td
(i)Tj
/T1_2 1 Tf
0.622 0 Td
( )Tj
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1.278 0 Td
[(0)-250(to)]TJ
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1.889 0 Td
(n)Tj
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0.822 0 Td
(\000)Tj
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(1)Tj
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(do)Tj
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7.9701 0 0 7.9701 323.2565 631.9012 Tm
(2:)Tj
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9.9626 0 0 9.9626 349.3825 631.9012 Tm
(t)Tj
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0.639 0 Td
[( )-278(\016)]TJ
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(\(\()Tj
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2.701 0.231 Td
([)Tj
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-0.146 -0.231 Td
(b)Tj
/T1_2 1 Tf
0.651 0 Td
(\000)Tj
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1 0 Td
(a)Tj
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(\))Tj
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1.14 0 Td
(\001)Tj
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0.631 0 Td
(b)Tj
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(v)Tj
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(i)Tj
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(\))Tj
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(3:)Tj
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(b)Tj
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(x)Tj
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(i)Tj
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9.9626 0 0 9.9626 361.1605 619.946 Tm
[( )-278(\016)]TJ
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(\()Tj
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(t)Tj
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2.75 0 Td
(+)Tj
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(b)Tj
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(a)Tj
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(\))Tj
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(4:)Tj
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(b)Tj
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(f)Tj
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6.9738 0 0 6.9738 354.26 605.1404 Tm
(i)Tj
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9.9626 0 0 9.9626 360.3442 606.6348 Tm
[( )-278(\016)]TJ
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(\()Tj
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(f)Tj
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2.764 0 Td
(\()Tj
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0.563 0 Td
(b)Tj
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(x)Tj
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(i)Tj
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(\)\))Tj
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(5:)Tj
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(b)Tj
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(y)Tj
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6.9738 0 0 6.9738 354.2669 591.8292 Tm
(i)Tj
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9.9626 0 0 9.9626 360.3511 593.3236 Tm
[( )-278(\016)]TJ
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(\()Tj
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2.3 0.264 Td
(b)Tj
/T1_1 1 Tf
-0.133 -0.264 Td
(f)Tj
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(i)Tj
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(\001)Tj
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(c)Tj
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-0.024 0 Td
(w)Tj
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(i)Tj
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(\))Tj
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(6:)Tj
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9.9626 0 0 9.9626 334.4385 581.3684 Tm
[(end)-250(f)25(or)]TJ
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7.9701 0 0 7.9701 323.2565 569.4133 Tm
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/T1_5 1 Tf
9.9626 0 0 9.9626 169.753 665.1381 Tm
(\))Tj
/T1_4 1 Tf
(j)Tj
/T1_5 1 Tf
0.944 0 Td
(=)Tj
/T1_4 1 Tf
1.056 0 Td
(j)Tj
/T1_2 1 Tf
(f)Tj
/T1_6 1 Tf
6.9738 0 0 6.9738 198.3955 669.2516 Tm
(0)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 201.1906 665.1381 Tm
(\()Tj
/T1_2 1 Tf
(\022)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 209.7419 663.6437 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 213.0587 665.1381 Tm
(\)\()Tj
/T1_1 1 Tf
0.814 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 226.5014 663.6437 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 232.0321 665.1381 Tm
(\000)Tj
/T1_2 1 Tf
1 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 247.6886 663.6437 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 251.0055 665.1381 Tm
(\))Tj
/T1_4 1 Tf
(j)Tj
/T1_2 1 Tf
(;)Tj
/T1_0 1 Tf
-17.298 -2.107 Td
[(for)-297(some)]TJ
/T1_2 1 Tf
3.87 0 Td
(\022)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 121.9122 642.6548 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 128.8571 644.1492 Tm
(2)Tj
/T1_5 1 Tf
1.031 0 Td
[([min)1(\()]TJ
/T1_2 1 Tf
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 168.067 642.6548 Tm
(i)Tj
/T1_2 1 Tf
9.9626 0 0 9.9626 171.3838 644.1492 Tm
(;)Tj
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0.48 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 181.5056 642.6548 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 184.8224 644.1492 Tm
(\))Tj
/T1_2 1 Tf
(;)Tj
/T1_5 1 Tf
0.833 0 Td
(max\()Tj
/T1_2 1 Tf
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 221.2345 642.6548 Tm
(i)Tj
/T1_2 1 Tf
9.9626 0 0 9.9626 224.5514 644.1492 Tm
(;)Tj
/T1_1 1 Tf
0.48 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 234.6731 642.6548 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 237.99 644.1492 Tm
(\)])Tj
/T1_0 1 Tf
0.963 0 Td
[(and)-297(with)-296(an)]TJ
-16.954 -1.2 Td
[(upper)-258(bound)-258(on)]TJ
/T1_2 1 Tf
6.551 0 Td
(f)Tj
/T1_6 1 Tf
6.9738 0 0 6.9738 149.8889 635.8094 Tm
(0)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 155.2532 632.194 Tm
[(we)-258(can)-258(bound)-258(this)-258(e)1(rror)-258(absolutely)65(.)]TJ
-7.686 -1.303 Td
(Let)Tj
/T1_1 1 Tf
1.817 0.264 Td
(b)Tj
/T1_2 1 Tf
-0.187 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 99.7841 617.7223 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 106.7175 619.2167 Tm
(=)Tj
/T1_4 1 Tf
1.141 0 Td
(\016)Tj
/T1_5 1 Tf
(\()Tj
/T1_2 1 Tf
(f)Tj
/T1_5 1 Tf
1.486 0 Td
(\()Tj
/T1_1 1 Tf
0.425 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 142.4567 617.7223 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 145.7736 619.2167 Tm
(\)\))Tj
/T1_0 1 Tf
1.074 0 Td
[(be)-296(the)-296(\003oating-point)-296(number)-296(com-)]TJ
-7.809 -1.2 Td
[(puted.)-310(At)-250(this)-250(step)-250(we)-250(no)25(w)-250(ha)20(v)15(e:)]TJ
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0.244 -2.209 Td
(\016)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 86.7999 583.6453 Tm
(b)Tj
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-0.181 -0.264 Td
(f)Tj
/T1_8 1 Tf
4.9813 0 0 4.9813 89.4197 580.8088 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 95.8412 585.2504 Tm
(=)Tj
/T1_4 1 Tf
1.056 0 Td
(j)Tj
/T1_1 1 Tf
0.465 0.264 Td
(b)Tj
/T1_2 1 Tf
-0.187 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 114.0023 583.756 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 119.533 585.2504 Tm
(\000)Tj
/T1_2 1 Tf
1 0 Td
(f)Tj
/T1_5 1 Tf
0.597 0 Td
(\()Tj
/T1_2 1 Tf
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 145.0138 583.756 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 148.3307 585.2504 Tm
(\))Tj
/T1_4 1 Tf
[(j)-1000(\024)-1000(j)]TJ
/T1_2 1 Tf
3.722 0 Td
(f)Tj
/T1_6 1 Tf
6.9738 0 0 6.9738 191.3638 589.3639 Tm
(0)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 194.1589 585.2504 Tm
(\()Tj
/T1_2 1 Tf
(\022)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 202.7102 583.756 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 206.027 585.2504 Tm
(\)\()Tj
/T1_1 1 Tf
0.814 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 219.4697 583.756 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 225.0004 585.2504 Tm
(\000)Tj
/T1_2 1 Tf
1 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 240.6569 583.756 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 243.9738 585.2504 Tm
(\))Tj
/T1_4 1 Tf
(j)Tj
/T1_5 1 Tf
0.889 0 Td
[(+)-222(ul)1(p\()]TJ
/T1_1 1 Tf
2.965 0.264 Td
(b)Tj
/T1_2 1 Tf
-0.188 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 285.381 583.756 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 288.6978 585.2504 Tm
(\))Tj
/T1_4 1 Tf
-12.423 -2.203 Td
(\024)Tj
/T1_5 1 Tf
1.898 0.677 Td
(17)Tj
ET
q
1 0 0 1 183.8419189 565.5953064 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
9.963 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 186.3326 556.47 Tm
(4)Tj
/T1_2 1 Tf
0.87 0.686 Td
(M)Tj
/T1_9 1 Tf
6.9738 0 0 6.9738 204.6653 561.8095 Tm
(1)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 211.3485 563.3039 Tm
(\001)Tj
/T1_5 1 Tf
0.5 0 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.814 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 239.7351 561.8095 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 243.052 563.3039 Tm
[(\))-222(+)-222(ul)1(p\()]TJ
/T1_1 1 Tf
3.576 0.264 Td
(b)Tj
/T1_2 1 Tf
-0.188 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 281.6918 561.8095 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 285.0086 563.3039 Tm
(\))Tj
/T1_2 1 Tf
(:)Tj
/T1_0 1 Tf
-21.961 -3.988 Td
[(2.)-500(Computation)-255(of)-255(the)]TJ
/T1_2 1 Tf
9.293 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 163.6875 522.0774 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 169.862 523.5718 Tm
(=)Tj
/T1_2 1 Tf
1.065 0 Td
(f)Tj
/T1_5 1 Tf
0.597 0 Td
(\()Tj
/T1_2 1 Tf
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 195.9867 522.0774 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 199.3036 523.5718 Tm
(\))Tj
/T1_4 1 Tf
0.615 0 Td
(\001)Tj
/T1_2 1 Tf
0.504 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 217.5778 522.0774 Tm
(i)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 220.8947 523.5718 Tm
[(.)-325(The)-255(accumulated)]TJ
-14.275 -1.2 Td
[(error)-250(so)-250(f)10(ar:)]TJ
/T1_4 1 Tf
0.769 -2.338 Td
(j)Tj
/T1_1 1 Tf
0.319 0 Td
(b)Tj
/T1_2 1 Tf
-0.041 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 93.994 486.8326 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 99.5247 488.327 Tm
(\000)Tj
/T1_2 1 Tf
1 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 114.3718 486.8326 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 117.6886 488.327 Tm
[(j)-1000(\024)]TJ
/T1_5 1 Tf
3.176 0.677 Td
(1)Tj
ET
q
1 0 0 1 149.3256073 490.6183014 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 149.3256 481.493 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.912 0 Td
(b)Tj
/T1_2 1 Tf
-0.134 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 178.0984 486.8325 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 181.4152 488.3269 Tm
[(\))-222(+)]TJ
/T1_4 1 Tf
1.611 0 Td
(j)Tj
/T1_1 1 Tf
0.411 0.264 Td
(b)Tj
/T1_2 1 Tf
-0.133 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 205.1111 486.8325 Tm
(i)Tj
/T1_1 1 Tf
9.9626 0 0 9.9626 208.6712 488.3269 Tm
(c)Tj
/T1_2 1 Tf
-0.024 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 215.5603 486.8325 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 221.091 488.3269 Tm
(\000)Tj
/T1_2 1 Tf
1 0 Td
(f)Tj
/T1_5 1 Tf
0.597 0 Td
(\()Tj
/T1_2 1 Tf
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 246.5718 486.8325 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 249.8887 488.3269 Tm
(\))Tj
/T1_2 1 Tf
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 260.8955 486.8325 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 264.2123 488.3269 Tm
(j)Tj
-13.43 -2.407 Td
(\024)Tj
/T1_5 1 Tf
1.898 0.677 Td
(1)Tj
ET
q
1 0 0 1 149.3256073 466.6392059 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 149.3256 457.5139 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.912 0 Td
(b)Tj
/T1_2 1 Tf
-0.134 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 178.0984 462.8534 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 181.4152 464.3478 Tm
[(\))-222(+)]TJ
/T1_1 1 Tf
-3.208 -1.808 Td
(b)Tj
/T1_2 1 Tf
-0.133 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 153.0077 442.2102 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 156.3245 443.7046 Tm
(j)Tj
/T1_1 1 Tf
0.302 0 Td
(c)Tj
/T1_2 1 Tf
-0.024 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 166.2243 442.2102 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 171.755 443.7046 Tm
(\000)Tj
/T1_2 1 Tf
1 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 188.85 442.2102 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 192.1669 443.7046 Tm
(j)Tj
/T1_5 1 Tf
0.5 0 Td
(+)Tj
/T1_2 1 Tf
1 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 214.2432 442.2102 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 217.56 443.7046 Tm
(j)Tj
/T1_1 1 Tf
0.411 0.264 Td
(b)Tj
/T1_2 1 Tf
-0.133 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 225.205 442.2102 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 230.7357 443.7046 Tm
(\000)Tj
/T1_2 1 Tf
1 0 Td
(f)Tj
/T1_5 1 Tf
0.597 0 Td
(\()Tj
/T1_2 1 Tf
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 256.2166 442.2102 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 259.5334 443.7046 Tm
(\))Tj
/T1_4 1 Tf
(j)Tj
-12.96 -1.971 Td
(\024)Tj
/T1_5 1 Tf
1.898 0.677 Td
(1)Tj
ET
q
1 0 0 1 149.3256073 426.3601074 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 149.3256 417.2348 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.912 0 Td
(b)Tj
/T1_2 1 Tf
-0.134 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 178.0984 422.5743 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 181.4152 424.0687 Tm
[(\))-222(+)]TJ
1.731 0.677 Td
(1)Tj
ET
q
1 0 0 1 198.6616211 426.3601074 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 198.6616 417.2348 Tm
(2)Tj
/T1_1 1 Tf
0.753 0.95 Td
(b)Tj
/T1_2 1 Tf
-0.133 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 209.716 422.5743 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 213.0329 424.0687 Tm
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.802 0 Td
(c)Tj
/T1_2 1 Tf
-0.024 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 237.8767 422.5743 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 241.1936 424.0687 Tm
[(\))-222(+)]TJ
/T1_2 1 Tf
1.611 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 264.3768 422.5743 Tm
(i)Tj
/T1_2 1 Tf
9.9626 0 0 9.9626 267.6937 424.0687 Tm
(\016)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 273.3865 422.4636 Tm
(b)Tj
/T1_3 1 Tf
-0.181 -0.264 Td
(f)Tj
/T1_8 1 Tf
4.9813 0 0 4.9813 276.0063 419.627 Tm
(i)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 130.4188 400.0896 Tm
(\024)Tj
/T1_5 1 Tf
1.898 0.677 Td
(3)Tj
ET
q
1 0 0 1 149.3256073 402.3809967 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 149.3256 393.2557 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.912 0 Td
(b)Tj
/T1_2 1 Tf
-0.134 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 178.0984 398.5952 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 181.4152 400.0896 Tm
[(\))-222(+)]TJ
/T1_2 1 Tf
-3.341 -2.536 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 155.2625 373.3303 Tm
(i)Tj
/T1_1 1 Tf
9.9626 0 0 9.9626 160.2398 388.8721 Tm
(\024)Tj
/T1_5 1 Tf
0.648 -0.733 Td
(17)Tj
ET
q
1 0 0 1 166.6934204 377.1161041 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
9.963 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 169.184 367.9908 Tm
(4)Tj
/T1_2 1 Tf
0.87 0.686 Td
(M)Tj
/T1_9 1 Tf
6.9738 0 0 6.9738 187.5167 373.3303 Tm
(1)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 194.2 374.8247 Tm
(\001)Tj
/T1_5 1 Tf
0.5 0 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.814 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 222.5865 373.3303 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 225.9034 374.8247 Tm
[(\))-222(+)-222(ul)1(p\()]TJ
/T1_1 1 Tf
3.576 0.264 Td
(b)Tj
/T1_2 1 Tf
-0.188 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 264.5432 373.3303 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 267.86 374.8247 Tm
(\))Tj
/T1_1 1 Tf
0.389 1.41 Td
(\025)Tj
/T1_0 1 Tf
-11.407 -3.438 Td
[([Lemmas)-250(1)-250(and)-250(2])]TJ
/T1_4 1 Tf
-2.778 -1.881 Td
(\024)Tj
/T1_5 1 Tf
1.898 0.677 Td
(3)Tj
ET
q
1 0 0 1 149.3256073 338.1676941 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 149.3256 329.0424 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.912 0 Td
(b)Tj
/T1_2 1 Tf
-0.134 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 178.0984 334.3819 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 181.4152 335.8763 Tm
[(\))-222(+)]TJ
-3.341 -2.407 Td
[(\(1)-222(+)-222(2)]TJ
/T1_6 1 Tf
6.9738 0 0 6.9738 174.1436 316.0107 Tm
(\000)Tj
/T1_3 1 Tf
(p)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 184.9754 311.8972 Tm
(\))Tj
/T1_1 1 Tf
0.413 0 Td
(c)Tj
/T1_2 1 Tf
-0.024 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 195.9822 310.4028 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 200.4946 318.637 Tm
(17)Tj
ET
q
1 0 0 1 200.4945221 314.1885986 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
9.963 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 202.9852 305.0633 Tm
(4)Tj
/T1_2 1 Tf
0.87 0.686 Td
(M)Tj
/T1_9 1 Tf
6.9738 0 0 6.9738 221.3179 310.4028 Tm
(1)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 228.0011 311.8972 Tm
(\001)Tj
/T1_5 1 Tf
0.5 0 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.814 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 256.3877 310.4028 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 259.7046 311.8972 Tm
[(\))-222(+)]TJ
-11.199 -2.072 Td
[(\(1)-222(+)-222(2)]TJ
/T1_6 1 Tf
6.9738 0 0 6.9738 174.1436 295.3675 Tm
(\000)Tj
/T1_3 1 Tf
(p)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 184.9754 291.254 Tm
(\))Tj
/T1_1 1 Tf
0.413 0 Td
(c)Tj
/T1_2 1 Tf
-0.024 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 195.9822 289.7596 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 199.299 291.254 Tm
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.965 0.264 Td
(b)Tj
/T1_2 1 Tf
-0.187 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 221.888 289.7596 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 225.2049 291.254 Tm
(\))Tj
/T1_0 1 Tf
-6.736 -1.5 Td
[([Lemma)-250(3])]TJ
/T1_4 1 Tf
-2.778 -1.881 Td
(\024)Tj
/T1_5 1 Tf
1.778 0 Td
(\()Tj
0.509 0.677 Td
(7)Tj
ET
q
1 0 0 1 153.2000122 259.8572998 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 153.2 250.732 Tm
(2)Tj
0.842 0.686 Td
[(+)-222(2)]TJ
/T1_9 1 Tf
6.9738 0 0 6.9738 176.5346 261.6794 Tm
(1)Tj
/T1_6 1 Tf
(\000)Tj
/T1_3 1 Tf
(p)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 191.3376 257.5659 Tm
[(\)u)1(lp\()]TJ
/T1_1 1 Tf
2.301 0 Td
(b)Tj
/T1_2 1 Tf
-0.134 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 217.8079 256.0715 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 221.1247 257.5659 Tm
[(\))-222(+)]TJ
-7.327 -2.407 Td
[(\(1)-222(+)-222(2)]TJ
/T1_6 1 Tf
6.9738 0 0 6.9738 174.1436 237.7003 Tm
(\000)Tj
/T1_3 1 Tf
(p)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 184.9754 233.5868 Tm
(\))Tj
/T1_2 1 Tf
(M)Tj
/T1_9 1 Tf
6.9738 0 0 6.9738 198.5149 232.0924 Tm
(1)Tj
/T1_1 1 Tf
9.9626 0 0 9.9626 203.2276 233.5868 Tm
(c)Tj
/T1_2 1 Tf
-0.024 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 210.1167 232.0924 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 214.6291 240.3266 Tm
(17)Tj
ET
q
1 0 0 1 214.6290131 235.8782043 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
9.963 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 217.1197 226.7529 Tm
(4)Tj
0.87 0.686 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.814 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 249.1925 232.0924 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 252.5094 233.5868 Tm
(\))Tj
/T1_0 1 Tf
-9.477 -1.764 Td
[([Lemmas)-250(1)-250(and)-250(2])]TJ
/T1_4 1 Tf
-2.778 -1.881 Td
(\024)Tj
/T1_5 1 Tf
1.778 0 Td
(\()Tj
0.509 0.677 Td
(7)Tj
ET
q
1 0 0 1 153.2000122 199.5605011 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 153.2 190.4352 Tm
(2)Tj
0.842 0.686 Td
[(+)-222(2)]TJ
/T1_9 1 Tf
6.9738 0 0 6.9738 176.5346 201.3826 Tm
(1)Tj
/T1_6 1 Tf
(\000)Tj
/T1_3 1 Tf
(p)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 191.3376 197.2691 Tm
[(\)u)1(lp\()]TJ
/T1_1 1 Tf
2.301 0 Td
(b)Tj
/T1_2 1 Tf
-0.134 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 217.8079 195.7747 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 221.1247 197.2691 Tm
[(\))-222(+)]TJ
/T1_1 1 Tf
-7.327 -1.126 Td
(\022)Tj
/T1_5 1 Tf
0.856 -0.734 Td
(17)Tj
ET
q
1 0 0 1 156.659317 174.2955933 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
9.963 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 159.1499 165.1703 Tm
(4)Tj
1.092 0.686 Td
[(+)-222(17)]TJ
/T1_4 1 Tf
2.222 0 Td
(\001)Tj
/T1_5 1 Tf
0.5 0 Td
(2)Tj
/T1_6 1 Tf
6.9738 0 0 6.9738 202.1331 176.1177 Tm
(\000)Tj
/T1_3 1 Tf
(p)Tj
/T1_6 1 Tf
(\000)Tj
/T1_9 1 Tf
(2)Tj
/T1_1 1 Tf
9.9626 0 0 9.9626 223.1627 186.0517 Tm
(\023)Tj
/T1_2 1 Tf
0.903 -1.41 Td
(M)Tj
/T1_9 1 Tf
6.9738 0 0 6.9738 241.8219 170.5098 Tm
(1)Tj
/T1_1 1 Tf
9.9626 0 0 9.9626 246.5346 172.0042 Tm
(c)Tj
/T1_2 1 Tf
-0.024 0 Td
(w)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 253.4237 170.5098 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 256.7405 172.0042 Tm
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.814 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 280.1459 170.5098 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 283.4627 172.0042 Tm
(\))Tj
-15.362 -2.044 Td
(=)Tj
/T1_2 1 Tf
1.778 0 Td
(\016)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 153.9007 150.0309 Tm
(b)Tj
/T1_3 1 Tf
-0.193 0 Td
(y)Tj
/T1_8 1 Tf
4.9813 0 0 4.9813 156.5984 149.0347 Tm
(i)Tj
/T1_2 1 Tf
9.9626 0 0 9.9626 160.2525 151.636 Tm
(:)Tj
/T1_0 1 Tf
-8.188 -2.29 Td
[(Remark:)-352(when)-271(bounding)-271(the)-271(error)-271(on)]TJ
/T1_1 1 Tf
15.295 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 236.3971 127.3263 Tm
(i)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 239.714 128.8207 Tm
(,)Tj
/T1_1 1 Tf
0.714 0.264 Td
(b)Tj
/T1_2 1 Tf
-0.188 -0.264 Td
(f)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 249.8339 127.3263 Tm
(i)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 255.8502 128.8207 Tm
[(as)-271(well)-271(as)]TJ
/T1_1 1 Tf
-17.743 -1.2 Td
(b)Tj
/T1_2 1 Tf
-0.041 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 83.5617 115.3711 Tm
(i)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 86.8786 116.8655 Tm
[(,)-274(the)-270(term)-270(wit)1(h)]TJ
/T1_5 1 Tf
6.166 0 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.814 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 171.7138 115.3711 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 175.0307 116.8655 Tm
(\))Tj
/T1_0 1 Tf
0.658 0 Td
[(v)25(anishes)-270(if)-269(the)-270(error)-269(on)]TJ
/T1_1 1 Tf
9.578 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 282.3547 115.3711 Tm
(i)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 288.357 116.8655 Tm
(is)Tj
-21.047 -1.2 Td
[(zero.)-623(One)-354(can)-354(easily)-355(sho)25(w)-354(with)-354(our)-354(assumption)-355(that)]TJ
T*
[(no)-280(under\003o)25(w)-281(occurs,)-288(and)-280(that)-281(if)]TJ
/T1_1 1 Tf
13.171 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 215.2302 91.4608 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 221.8755 92.9552 Tm
[(=)-334(0)]TJ
/T1_0 1 Tf
1.892 0 Td
[(then)-280(the)-281(error)]TJ
-16.266 -1.2 Td
(on)Tj
/T1_1 1 Tf
1.304 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 97.0094 79.5056 Tm
(i)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 103.0019 81 Tm
[(is)-269(zero)-268(\(i.e.,)]TJ
/T1_2 1 Tf
5.003 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 158.5418 79.5056 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 164.9686 81 Tm
[(=)-312(0)]TJ
/T1_0 1 Tf
[(\))-269(and)-268(the)-269(ill-de\002ned)-268(quantity)]TJ
/T1_5 1 Tf
17.311 63.14 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.814 0 Td
(b)Tj
/T1_2 1 Tf
-0.036 0 Td
(x)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 360.8325 708.543 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 364.1494 710.0374 Tm
(\))Tj
/T1_0 1 Tf
0.609 0 Td
[(v)25(anishes.)-300(F)15(or)-220(the)-220(error)-219(bound)-220(we)-220(k)10(eep)-220(track)-220(of)]TJ
-3.291 -1.2 Td
(only)Tj
/T1_5 1 Tf
2.028 0 Td
(max\()Tj
/T1_2 1 Tf
(\016)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 384.794 696.4771 Tm
(b)Tj
/T1_3 1 Tf
-0.046 0 Td
(y)Tj
/T1_8 1 Tf
4.9813 0 0 4.9813 388.5156 695.4808 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 392.1697 698.0822 Tm
(\))Tj
/T1_0 1 Tf
(.)Tj
-6.745 -2.155 Td
[(3.)-500(Summation)-275(of)-274(the)]TJ
/T1_2 1 Tf
8.741 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 416.9409 675.1136 Tm
(i)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 420.2577 676.608 Tm
[(')55(s:)-359(this)-275(is)-275(done)-274(with)-275(Demmel)-275(and)]TJ
-8.314 -1.2 Td
[(Hida)-387(summation)-387(algorithm)-387([3],)-421(which)-387(guarantees)-387(an)]TJ
T*
[(error)-288(of)-287(at)-288(most)]TJ
/T1_5 1 Tf
6.593 0 Td
(1)Tj
/T1_2 1 Tf
(:)Tj
/T1_5 1 Tf
[(5)-288(u)1(lp)]TJ
/T1_0 1 Tf
3.242 0 Td
[(on)-288(the)-287(\002nal)-288(result.)-423(This)-287(algo-)]TJ
-9.835 -1.2 Td
[(rithm)-203(uses)-203(a)-202(lar)18(ger)-203(w)10(orking)-203(precision)]TJ
/T1_2 1 Tf
14.853 0 Td
(p)Tj
/T1_6 1 Tf
6.9738 0 0 6.9738 490.4109 644.3578 Tm
(0)Tj
/T1_4 1 Tf
9.9626 0 0 9.9626 495.9733 640.7425 Tm
(\031)Tj
/T1_2 1 Tf
1.056 0 Td
(p)Tj
/T1_5 1 Tf
0.551 0 Td
[(+)-47(log)]TJ
/T1_9 1 Tf
6.9738 0 0 6.9738 533.0632 638.3072 Tm
(2)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 537.5325 640.7425 Tm
(\()Tj
/T1_2 1 Tf
(n)Tj
/T1_5 1 Tf
(\))Tj
/T1_0 1 Tf
(.)Tj
-20.086 -1.304 Td
(Let)Tj
/T1_2 1 Tf
1.583 0 Td
(S)Tj
/T1_5 1 Tf
0.949 0 Td
(=)Tj
/T1_1 1 Tf
1.056 0.75 Td
(P)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 383.6807 632.7613 Tm
(n)Tj
/T1_6 1 Tf
(\000)Tj
/T1_9 1 Tf
(1)Tj
/T1_3 1 Tf
0 -1.147 TD
(i)Tj
/T1_9 1 Tf
(=0)Tj
/T1_2 1 Tf
9.9626 0 0 9.9626 400.9619 627.7522 Tm
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 405.8464 626.2578 Tm
(i)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 409.1632 627.7522 Tm
(.)Tj
/T1_4 1 Tf
-3.698 -2.799 Td
(j)Tj
/T1_1 1 Tf
0.419 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.141 -0.253 Td
(S)Tj
/T1_4 1 Tf
0.893 0 Td
(\000)Tj
/T1_2 1 Tf
1 0 Td
(S)Tj
/T1_4 1 Tf
0.671 0 Td
[(j)-278(\024)]TJ
/T1_5 1 Tf
1.731 0.677 Td
(3)Tj
ET
q
1 0 0 1 417.8791199 602.1611938 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 417.8791 593.0359 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
/T1_1 1 Tf
1.919 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.141 -0.253 Td
(S)Tj
/T1_5 1 Tf
0.671 0 Td
[(\))-222(+)]TJ
/T1_2 1 Tf
1.611 0 Td
(n)Tj
/T1_4 1 Tf
0.822 0 Td
(\001)Tj
/T1_5 1 Tf
0.5 0 Td
(max\()Tj
/T1_2 1 Tf
(\016)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 504.8393 598.2647 Tm
(b)Tj
/T1_3 1 Tf
-0.046 0 Td
(y)Tj
/T1_8 1 Tf
4.9813 0 0 4.9813 508.5609 597.2684 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 512.215 599.8698 Tm
(\))Tj
/T1_2 1 Tf
(:)Tj
/T1_0 1 Tf
-18.794 -3.221 Td
[(4.)-500(Multiplication)-259(by)]TJ
/T1_3 1 Tf
6.9738 0 0 6.9738 410.7644 571.7 Tm
(b)Tj
/T1_6 1 Tf
(\000)Tj
/T1_3 1 Tf
(a)Tj
ET
q
1 0 0 1 410.7644196 570.0688019 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
14.052 0.199 l
S
Q
BT
/T1_9 1 Tf
6.9738 0 0 6.9738 415.8046 564.3419 Tm
(2)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 426.0115 567.7774 Tm
(:)Tj
/T1_2 1 Tf
0.606 0 Td
(I)Tj
/T1_5 1 Tf
0.813 0 Td
(=)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 452.0297 571.7 Tm
(b)Tj
/T1_6 1 Tf
(\000)Tj
/T1_3 1 Tf
(a)Tj
ET
q
1 0 0 1 452.0297089 570.0688019 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
14.052 0.199 l
S
Q
BT
/T1_9 1 Tf
6.9738 0 0 6.9738 457.0699 564.3419 Tm
(2)Tj
/T1_2 1 Tf
9.9626 0 0 9.9626 467.2768 567.7774 Tm
(S)Tj
/T1_0 1 Tf
0.671 0 Td
[(.)-337(W)80(e)-260(note)]TJ
/T1_2 1 Tf
4.136 0 Td
(D)Tj
/T1_5 1 Tf
1.15 0 Td
(=)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 538.5048 571.7 Tm
(b)Tj
/T1_6 1 Tf
(\000)Tj
/T1_3 1 Tf
(a)Tj
ET
q
1 0 0 1 538.5048218 570.0688019 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
14.052 0.199 l
S
Q
BT
/T1_9 1 Tf
6.9738 0 0 6.9738 543.545 564.3419 Tm
(2)Tj
/T1_0 1 Tf
9.9626 0 0 9.9626 337.4273 552.8202 Tm
[(and)-376(assume)-375(as)-376(before)-376(that)-375(the)-376(input)]TJ
/T1_10 1 Tf
15.328 0.231 Td
([)Tj
/T1_2 1 Tf
-0.146 -0.231 Td
(b)Tj
/T1_4 1 Tf
0.651 0 Td
(\000)Tj
/T1_2 1 Tf
1 0 Td
(a)Tj
/T1_0 1 Tf
0.904 0 Td
[(w)10(as)-376(com-)]TJ
-17.738 -1.2 Td
[(puted)-197(as)-197(the)-196(rounded)-197(to)-197(nearest)-197(of)-197(its)-197(e)15(xact)-196(v)25(alue.)-293(Since)]TJ
0 -1.2 TD
[(the)-250(di)25(vision)-250(by)]TJ
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6.17 0 Td
(2)Tj
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0.75 0 Td
[(is)-250(e)15(xact)-250(in)-250(binary)-250(we)-250(ha)20(v)15(e:)]TJ
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-6.613 -2.715 Td
(j)Tj
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0.483 0.253 Td
(b)Tj
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-0.206 -0.253 Td
(D)Tj
/T1_4 1 Tf
1.078 0 Td
(\000)Tj
/T1_2 1 Tf
1 0 Td
(D)Tj
/T1_4 1 Tf
0.856 0 Td
[(j)-1000(\024)]TJ
/T1_5 1 Tf
3.176 0.677 Td
(1)Tj
ET
q
1 0 0 1 404.1146088 504.1508942 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 404.1146 495.0256 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
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1.983 0.253 Td
(b)Tj
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-0.206 -0.253 Td
(D)Tj
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0.856 0 Td
(\))Tj
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-8.965 -2.407 Td
(j)Tj
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0.37 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.092 -0.253 Td
(I)Tj
/T1_4 1 Tf
0.74 0 Td
(\000)Tj
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1 0 Td
(I)Tj
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0.518 0 Td
[(j)-1000(\024)]TJ
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3.176 0.677 Td
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ET
q
1 0 0 1 404.1146088 480.1717987 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
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S
Q
BT
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9.9626 0 0 9.9626 404.1146 471.0465 Tm
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0.62 0.686 Td
[(ul)1(p\()]TJ
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(b)Tj
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-0.092 -0.253 Td
(I)Tj
/T1_5 1 Tf
0.518 0 Td
[(\))-222(+)]TJ
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1.611 0 Td
(j)Tj
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0.419 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.141 -0.253 Td
(S)Tj
/T1_1 1 Tf
0.876 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.206 -0.253 Td
(D)Tj
/T1_4 1 Tf
1.078 0 Td
(\000)Tj
/T1_2 1 Tf
1 0 Td
[(S)-58(D)]TJ
/T1_4 1 Tf
1.527 0 Td
(j)Tj
-10.978 -2.407 Td
(\024)Tj
/T1_5 1 Tf
1.898 0.676 Td
(1)Tj
ET
q
1 0 0 1 404.1146088 456.1927032 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 404.1146 447.0674 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
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1.87 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.092 -0.253 Td
(I)Tj
/T1_5 1 Tf
0.518 0 Td
[(\))-222(+)]TJ
1.731 0.677 Td
(1)Tj
ET
q
1 0 0 1 450.4104156 456.1927032 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
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S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 450.4104 447.0674 Tm
(2)Tj
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0.62 0.686 Td
(j)Tj
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0.419 0.253 Td
(b)Tj
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-0.141 -0.253 Td
(S)Tj
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0.671 0 Td
(j)Tj
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[(ul)1(p\()]TJ
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2.261 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.206 -0.253 Td
(D)Tj
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0.856 0 Td
[(\))-222(+)]TJ
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1.611 0 Td
(D)Tj
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0.856 0 Td
(j)Tj
/T1_1 1 Tf
0.419 0.253 Td
(b)Tj
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-0.141 -0.253 Td
(S)Tj
/T1_4 1 Tf
0.893 0 Td
(\000)Tj
/T1_2 1 Tf
1 0 Td
(S)Tj
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0.671 0 Td
(j)Tj
-16.333 -2.407 Td
(\024)Tj
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1.898 0.677 Td
(3)Tj
ET
q
1 0 0 1 404.1146088 432.2135925 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 404.1146 423.0883 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
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1.87 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.092 -0.253 Td
(I)Tj
/T1_5 1 Tf
0.518 0 Td
[(\))-222(+)]TJ
/T1_2 1 Tf
1.611 0 Td
(D)Tj
/T1_5 1 Tf
0.976 0.677 Td
(3)Tj
ET
q
1 0 0 1 458.9354095 432.2135925 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 458.9354 423.0883 Tm
(2)Tj
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0.62 0.686 Td
(D)Tj
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0.856 0 Td
[(ul)1(p\()]TJ
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1.919 0.253 Td
(b)Tj
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-0.141 -0.253 Td
(S)Tj
/T1_5 1 Tf
0.671 0 Td
[(\))-222(+)]TJ
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-9.547 -1.836 Td
(nD)Tj
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1.678 0 Td
(\001)Tj
/T1_5 1 Tf
0.5 0 Td
(max\()Tj
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(\016)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 451.7818 410.0262 Tm
(b)Tj
/T1_3 1 Tf
-0.046 0 Td
(y)Tj
/T1_8 1 Tf
4.9813 0 0 4.9813 455.5034 409.0299 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 459.1576 411.6313 Tm
(\))Tj
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-4.645 -1.5 Td
[([Lemmas)-250(1)-250(and)-250(2])]TJ
/T1_4 1 Tf
-2.778 -1.881 Td
(\024)Tj
/T1_5 1 Tf
1.898 0.677 Td
(3)Tj
ET
q
1 0 0 1 404.1146088 380.2346039 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 404.1146 371.1093 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
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1.87 0.253 Td
(b)Tj
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-0.092 -0.253 Td
(I)Tj
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0.518 0 Td
[(\))-222(+)-222(\(1)-222(+)-222(2)]TJ
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6.9738 0 0 6.9738 475.2285 382.0567 Tm
(\000)Tj
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(p)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 486.0602 377.9432 Tm
(\))Tj
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0.595 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.206 -0.253 Td
(D)Tj
/T1_5 1 Tf
0.976 0.677 Td
(3)Tj
ET
q
1 0 0 1 499.655014 380.2346039 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 499.655 371.1093 Tm
(2)Tj
0.62 0.686 Td
[(ul)1(p\()]TJ
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1.919 0.253 Td
(b)Tj
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-0.141 -0.253 Td
(S)Tj
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0.671 0 Td
[(\))-222(+)]TJ
-12.779 -2.061 Td
[(\(1)-222(+)-222(2)]TJ
/T1_6 1 Tf
6.9738 0 0 6.9738 428.9326 361.5242 Tm
(\000)Tj
/T1_3 1 Tf
(p)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 439.7644 357.4107 Tm
(\))Tj
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0.595 0.253 Td
(b)Tj
/T1_2 1 Tf
-0.206 -0.253 Td
[(D)-28(n)]TJ
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1.678 0 Td
(\001)Tj
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0.5 0 Td
(max\()Tj
/T1_2 1 Tf
(\016)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 492.5015 355.8056 Tm
(b)Tj
/T1_3 1 Tf
-0.046 0 Td
(y)Tj
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4.9813 0 0 4.9813 496.2231 354.8093 Tm
(i)Tj
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9.9626 0 0 9.9626 499.8772 357.4107 Tm
(\))Tj
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-8.732 -1.5 Td
[([Lemma)-250(3])]TJ
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-2.778 -1.881 Td
(\024)Tj
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1.778 0 Td
(\()Tj
0.509 0.677 Td
(9)Tj
ET
q
1 0 0 1 407.9890137 326.0140076 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
4.981 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 407.989 316.8887 Tm
(2)Tj
0.842 0.686 Td
[(+)-222(3)]TJ
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1.722 0 Td
(\001)Tj
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0.5 0 Td
(2)Tj
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6.9738 0 0 6.9738 443.5001 327.836 Tm
(\000)Tj
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(p)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 454.3319 323.7225 Tm
[(\)u)1(lp\()]TJ
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2.259 0.253 Td
(b)Tj
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-0.092 -0.253 Td
(I)Tj
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0.518 0 Td
[(\))-222(+)]TJ
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-7.845 -2.061 Td
(n)Tj
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[(\(1)-222(+)-222(2)]TJ
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6.9738 0 0 6.9738 434.9125 307.3035 Tm
(\000)Tj
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(p)Tj
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9.9626 0 0 9.9626 445.7443 303.19 Tm
(\))Tj
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0.595 0.253 Td
(b)Tj
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-0.206 -0.253 Td
(D)Tj
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1.078 0 Td
(\001)Tj
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0.5 0 Td
(max\()Tj
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(\016)Tj
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6.9738 0 0 6.9738 492.5015 301.5849 Tm
(b)Tj
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-0.046 0 Td
(y)Tj
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4.9813 0 0 4.9813 496.2231 300.5886 Tm
(i)Tj
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9.9626 0 0 9.9626 499.8772 303.19 Tm
(\))Tj
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(:)Tj
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-8.732 -1.5 Td
[([Lemmas)-250(1)-250(and)-250(2])]TJ
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-9.574 -4.255 Td
[(Cor)18(ollary)-331(1.)]TJ
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5.783 0 Td
[(If)-331(we)-331(assume)-331(furthermor)37(e)-331(that)-331(the)-331(sign)-331(of)]TJ
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17.333 0 Td
(f)Tj
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-23.116 -1.2 Td
[(does)-197(not)-196(c)15(hang)10(e)-197(on)]TJ
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7.761 0 Td
([)Tj
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[(a;)-166(b)]TJ
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(])Tj
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[(,)-207(then)-197(we)-197(have)-196(the)-197(following)-196(bound:)]TJ
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(B)Tj
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6.9738 0 0 6.9738 341.0047 210.6244 Tm
(0)Tj
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-0.072 -0.943 Td
(total)Tj
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9.9626 0 0 9.9626 356.5601 206.5109 Tm
(=)Tj
1.176 0.677 Td
(161)Tj
ET
q
1 0 0 1 368.2717133 208.8023071 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
14.944 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 373.2531 199.677 Tm
(4)Tj
1.12 0.686 Td
[(ul)1(p\()]TJ
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1.87 0.253 Td
(b)Tj
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0.518 0 Td
[(\))-222(+)]TJ
1.731 0.677 Td
(425)Tj
ET
q
1 0 0 1 424.5302124 208.8023071 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
14.944 0.199 l
S
Q
BT
/T1_5 1 Tf
9.9626 0 0 9.9626 427.0209 199.677 Tm
(64)Tj
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1.37 0.686 Td
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6.9738 0 0 6.9738 456.3148 205.0165 Tm
(1)Tj
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9.9626 0 0 9.9626 462.8327 209.0292 Tm
(b)Tj
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1.022 0 Td
(max\()Tj
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2.274 0 Td
(c)Tj
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(w)Tj
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6.9738 0 0 6.9738 500.518 205.0165 Tm
(i)Tj
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9.9626 0 0 9.9626 503.8348 206.5109 Tm
[(ul)1(p\()]TJ
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1.952 0 Td
(b)Tj
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-0.174 0 Td
(x)Tj
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6.9738 0 0 6.9738 527.2402 205.0165 Tm
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9.9626 0 0 9.9626 530.557 206.5109 Tm
(\)\))Tj
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1.12 0.677 Td
(\()Tj
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(b)Tj
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1.04 0 Td
(\000)Tj
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1 0 Td
(a)Tj
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(\))Tj
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6.9738 0 0 6.9738 406.5944 191.1977 Tm
(2)Tj
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(n)Tj
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(+1)Tj
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9.9626 0 0 9.9626 426.0759 187.5823 Tm
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(n)Tj
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(!\))Tj
/T1_9 1 Tf
6.9738 0 0 6.9738 442.5719 191.1977 Tm
(4)Tj
ET
q
1 0 0 1 377.1274109 183.1340027 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
69.914 0.199 l
S
Q
BT
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9.9626 0 0 9.9626 378.4097 174.0087 Tm
(\(2)Tj
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(n)Tj
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1.711 0 Td
[(+)-222(1\)[\(2)]TJ
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(n)Tj
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(\)!])Tj
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6.9738 0 0 6.9738 441.2898 176.8868 Tm
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9.9626 0 0 9.9626 448.2369 180.8425 Tm
(M)Tj
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6.9738 0 0 6.9738 457.902 179.3482 Tm
(2)Tj
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(n)Tj
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9.9626 0 0 9.9626 467.2963 180.8426 Tm
(:)Tj
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-15.011 -2.847 Td
(P)Tj
7.9701 0 0 7.9701 323.7883 152.4789 Tm
[(R)-22(O)-61(O)-62(F)]TJ
9.9626 0 0 9.9626 346.4431 152.4789 Tm
[(:)-481(Let)-336(us)-336(assume)-335(for)-336(e)15(xample)-336(that)]TJ
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13.979 0 Td
(f)Tj
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1.034 0 Td
(\025)Tj
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1.214 0 Td
(0)Tj
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[(,)-357(kno)25(wing)]TJ
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[(that)-250(the)-250(Gauss-Le)15(gendre)-250(weights)-250(are)-250(positi)25(v)15(e)-250(we)-250(ha)20(v)15(e)]TJ
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(8)Tj
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(i)Tj
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1.178 0 Td
(2)Tj
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0.944 0 Td
([0)Tj
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[(;)-166(n)]TJ
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(\000)Tj
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1 0 Td
(1])Tj
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(;)Tj
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1.356 0 Td
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6.9738 0 0 6.9738 433.1788 115.3148 Tm
(i)Tj
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9.9626 0 0 9.9626 439.263 116.8092 Tm
(=)Tj
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1.056 0 Td
(\016)Tj
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(\()Tj
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0.913 0 Td
(c)Tj
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-0.024 0 Td
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6.9738 0 0 6.9738 465.7672 115.3148 Tm
(i)Tj
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9.9626 0 0 9.9626 471.2979 116.8092 Tm
(\001)Tj
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0.633 0.264 Td
(b)Tj
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(f)Tj
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6.9738 0 0 6.9738 481.1568 115.3148 Tm
(i)Tj
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9.9626 0 0 9.9626 484.4736 116.8092 Tm
(\))Tj
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0.667 0 Td
(\025)Tj
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1.056 0 Td
(0)Tj
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-18.482 -2.278 Td
(so)Tj
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8.152 -1.317 Td
[(ul)1(p\()]TJ
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1.912 0 Td
(b)Tj
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-0.134 0 Td
(y)Tj
/T1_3 1 Tf
6.9738 0 0 6.9738 421.3111 79.5056 Tm
(i)Tj
/T1_5 1 Tf
9.9626 0 0 9.9626 424.6279 81 Tm
(\))Tj
/T1_4 1 Tf
0.667 0 Td
(\024)Tj
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1.056 0 Td
(2)Tj
/T1_9 1 Tf
6.9738 0 0 6.9738 446.7671 85.1135 Tm
(1)Tj
/T1_6 1 Tf
(\000)Tj
/T1_3 1 Tf
(p)Tj
/T1_1 1 Tf
9.9626 0 0 9.9626 462.9033 81 Tm
(b)Tj
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-0.134 0 Td
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[(\))-1000(=)-1000(l)1(og)]TJ
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[(Leibniz')55(s)-250(formula)-250(gi)25(v)15(es)]TJ
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[(\))-278(=)]TJ
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(=0)Tj
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(\022)Tj
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0.128 -1.362 Td
(i)Tj
/T1_3 1 Tf
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(\023)Tj
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1.142 -0.733 Td
(d)Tj
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ET
q
1 0 0 1 175.8209076 132.0296936 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
0 0.199 m
14.546 0.199 l
S
Q
BT
/T1_1 1 Tf
9.9626 0 0 9.9626 175.8209 122.9044 Tm
(d)Tj
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(x)Tj
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6.9738 0 0 6.9738 187.0496 125.7825 Tm
(i)Tj
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9.9626 0 0 9.9626 191.562 129.7383 Tm
(g)Tj
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0.513 0 Td
(\()Tj
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(x)Tj
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0.96 0 Td
(\))Tj
0.795 0.677 Td
(d)Tj
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6.9738 0 0 6.9738 219.6912 140.0935 Tm
(n)Tj
/T1_5 1 Tf
(\000)Tj
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(i)Tj
ET
q
1 0 0 1 211.3095093 132.0296936 cm
0 G
0.398 w 10 M 0 j 0 J []0 d
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S
Q
BT
/T1_1 1 Tf
9.9626 0 0 9.9626 211.3095 122.9044 Tm
(d)Tj
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(x)Tj
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6.9738 0 0 6.9738 222.5382 125.7825 Tm
(n)Tj
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(\000)Tj
/T1_4 1 Tf
(i)Tj
/T1_2 1 Tf
9.9626 0 0 9.9626 238.2021 129.7383 Tm
(h)Tj
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(x)Tj
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[(F)15(or)]TJ
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0.622 0 Td
(\025)Tj
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[(we)-250(can)-250(write)]TJ
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(\))Tj
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9.9626 0 0 9.9626 129.6588 81 Tm
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[(\))-278(=)-277(\()]TJ
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9.9626 0 0 9.9626 190.2678 81 Tm
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(i)Tj
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(\000)Tj
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(i)Tj
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9.9626 0 0 9.9626 236.6114 81 Tm
(:)Tj
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8.119 63.14 Td
[(The)-250(deri)25(v)25(ati)25(v)15(es)-250(of)]TJ
/T1_2 1 Tf
7.436 0 Td
(g)Tj
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[(need)-250(more)-250(w)10(ork,)-250(b)20(ut)-250(we)-250(can)-250(write)]TJ
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-0.645 -2.276 Td
(g)Tj
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6.9738 0 0 6.9738 397.8657 691.4751 Tm
(\()Tj
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(i)Tj
/T1_6 1 Tf
(\))Tj
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9.9626 0 0 9.9626 407.4092 687.3616 Tm
(\()Tj
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(x)Tj
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0.96 0 Td
[(\))-278(=)]TJ
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1.722 0 Td
(G)Tj
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6.9738 0 0 6.9738 441.9684 685.8672 Tm
(i)Tj
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9.9626 0 0 9.9626 445.2852 687.3616 Tm
(\()Tj
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(x)Tj
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0.96 0 Td
(\))Tj
/T1_2 1 Tf
(e)Tj
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(\000)Tj
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(x)Tj
/T1_9 1 Tf
4.9813 0 0 4.9813 474.1112 694.4816 Tm
(2)Tj
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9.9626 0 0 9.9626 317.502 665.9943 Tm
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(G)Tj
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6.9738 0 0 6.9738 352.1643 664.4999 Tm
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(x)Tj
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[(=)-1000(1)]TJ
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(G)Tj
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(i)Tj
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9.9626 0 0 9.9626 381.9473 629.683 Tm
(=)Tj
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(\000)Tj
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(2)Tj
/T1_2 1 Tf
(xG)Tj
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6.9738 0 0 6.9738 425.9158 628.1886 Tm
(i)Tj
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(x)Tj
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[(\))-222(+)]TJ
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(0)Tj
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[(and)-251(has)-252(only)-251(monomials)-251(of)-252(the)-251(same)-251(parity)-251(as)]TJ
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[(Furthermore)-250(the)-250(leading)-250(coef)25(\002cient)-250(of)]TJ
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[(W)80(e)-237(will)-236(no)25(w)-237(pro)15(v)15(e)-237(by)-236(recurrence)-237(that)-237(for)]TJ
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[(the)-237(coef-)]TJ
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[(\002cients)-250(of)]TJ
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(G)Tj
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(i)Tj
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[(are)-250(bounded)-250(in)-250(absolute)-250(v)25(alue)-250(by)]TJ
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[(+)-222(1\)!)]TJ
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[(The)-199(property)-198(is)-199(true)-198(for)]TJ
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(0)Tj
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(x)Tj
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[(\))-278(=)-277(1)]TJ
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[(.)-293(Assume)-198(the)-199(property)]TJ
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[(true)-250(for)-250(some)]TJ
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[(and)-250(write)]TJ
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[(\))-1000(=)]TJ
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(i)Tj
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[(\))-1000(=)]TJ
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(j)Tj
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(j)Tj
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[(F)15(or)]TJ
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(\024)Tj
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(\000)Tj
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0.75 0 Td
[(we)-250(ha)20(v)15(e)]TJ
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(j)Tj
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(=)Tj
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(\000)Tj
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(2)Tj
/T1_2 1 Tf
(a)Tj
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(j)Tj
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(\000)Tj
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(1)Tj
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[(+)-222(\()]TJ
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(j)Tj
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[(+)-222(1\))]TJ
/T1_2 1 Tf
(a)Tj
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(j)Tj
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0.531 0 Td
(+1)Tj
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(j)Tj
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(b)Tj
/T1_4 1 Tf
6.9738 0 0 6.9738 367.3541 401.5257 Tm
(j)Tj
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[(j)-1000(\024)]TJ
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(2\()Tj
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(i)Tj
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1.456 0 Td
[(+)-222(1\)!)-222(+)-222(\()]TJ
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(j)Tj
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[(+)-222(1\)\()]TJ
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(i)Tj
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[(+)-222(1\)!)]TJ
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(\024)Tj
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(j)Tj
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[(+)-222(3\)\()]TJ
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(i)Tj
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[(+)-222(1\)!)]TJ
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(\024)Tj
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(i)Tj
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[(+)-222(2\)!)]TJ
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(:)Tj
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(Since)Tj
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2.565 0 Td
(b)Tj
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6.9738 0 0 6.9738 347.3268 350.2705 Tm
(i)Tj
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[(=)-449(0)]TJ
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2.069 0 Td
(and)Tj
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1.787 0 Td
(j)Tj
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(b)Tj
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(i)Tj
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(j)Tj
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[(=)-449(2)]TJ
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(i)Tj
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(+1)Tj
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(<)Tj
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(\()Tj
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(i)Tj
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1.024 0 Td
[(+)-291(2\))1(!)]TJ
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2.578 0 Td
[(the)-343(property)]TJ
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[(holds)-250(for)]TJ
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3.833 0 Td
(i)Tj
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0.567 0 Td
[(+)-222(1)]TJ
/T1_0 1 Tf
(.)Tj
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[(F)15(or)]TJ
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1.624 0 Td
(n)Tj
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0.878 0 Td
(\025)Tj
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1.056 0 Td
(0)Tj
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0.75 0 Td
(and)Tj
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1.694 0 Td
(x)Tj
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(2)Tj
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([17)Tj
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(;)Tj
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1.722 0 Td
(42])Tj
/T1_0 1 Tf
1.528 0 Td
[(we)-250(kno)25(w)-250(that)]TJ
/T1_7 1 Tf
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(j)Tj
/T1_2 1 Tf
(G)Tj
/T1_4 1 Tf
6.9738 0 0 6.9738 393.1124 304.9928 Tm
(n)Tj
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9.9626 0 0 9.9626 398.5354 306.4872 Tm
(\()Tj
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(x)Tj
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0.96 0 Td
(\))Tj
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[(j)-278(\024)]TJ
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2 0 Td
(n)Tj
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0.822 0 Td
(\001)Tj
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0.5 0 Td
(\()Tj
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BT
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(54)Tj
ET
q
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q
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q
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q
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q
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q
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q
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q
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q
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q
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q
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q
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q
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q
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q
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q
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q
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BT
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q
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BT
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q
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BT
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q
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8.9664 0 0 8.9664 100.7964 632.3786 Tm
(32)Tj
ET
q
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(556)Tj
ET
q
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ET
q
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BT
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8.9664 0 0 8.9664 194.0676 632.3786 Tm
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ET
q
1 0 0 1 225.258316 629.0908966 cm
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BT
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ET
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Q
BT
/T1_1 1 Tf
8.9664 0 0 8.9664 269.8427 632.3786 Tm
(51)Tj
ET
q
1 0 0 1 288.6238098 629.0908966 cm
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q
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Q
BT
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9.9626 0 0 9.9626 70.7071 600.5031 Tm
[(Figure)-253(1.)]TJ
/T1_1 1 Tf
8.9664 0 0 8.9664 114.509 600.5031 Tm
[(Optimized)-228(order)]TJ
/T1_0 1 Tf
6.787 0 Td
(n)Tj
/T1_1 1 Tf
5.9776 0 0 5.9776 180.9236 599.5069 Tm
(opt)Tj
8.9664 0 0 8.9664 189.061 600.5031 Tm
[(,)-232(predicted)-228(precision)]TJ
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8.324 0 Td
(p)Tj
/T1_2 1 Tf
5.9776 0 0 5.9776 268.3244 599.4404 Tm
[(pre)-1(d)]TJ
/T1_1 1 Tf
8.9664 0 0 8.9664 70.7071 589.5442 Tm
[(and)-309(measured)-310(precision)]TJ
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9.87 0 Td
(p)Tj
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5.9776 0 0 5.9776 163.8327 588.548 Tm
(meas)Tj
/T1_1 1 Tf
8.9664 0 0 8.9664 182.9179 589.5442 Tm
[(for)-309(dif)25(ferent)-310(w)10(orking)-309(preci-)]TJ
-12.515 -1.222 Td
(sions)Tj
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2.327 0 Td
(p)Tj
/T1_1 1 Tf
0.787 0 Td
[(in)-271(bits)-271(and)-271(orders)]TJ
/T1_0 1 Tf
7.25 0 Td
(m)Tj
/T1_1 1 Tf
1.177 0 Td
[(of)-271(composition.)-373(The)-271(timings)]TJ
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11.77 0 Td
(t)Tj
/T1_1 1 Tf
-23.311 -1.222 Td
[(are)-323(gi)25(v)15(en)-324(in)-323(ms)-324(\(with)]TJ
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(\034)Tj
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[(the)-323(percentage)-324(of)-323(total)-324(time)-323(tak)10(en)]TJ
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[(by)-345(the)-345(weight)1(s)-345(computation\))-345(and)-345(were)-344(done)-345(on)-345(a)-345(2.4GHz)]TJ
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[(AMD)-250(Opteron\231)-250(250)-250(processor)55(.)]TJ
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(53)Tj
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[(bits)-222(and)]TJ
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3.333 0 Td
(113)Tj
/T1_1 1 Tf
1.722 0 Td
[(bits)-222(to)-222(reproduce)-221(the)-222(double)-222(and)-222(quad)-222(preci-)]TJ
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[(sion,)-293(and)-285(precisions)]TJ
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(200)Tj
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(,)Tj
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2.043 0 Td
(500)Tj
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(,)Tj
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(1000)Tj
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(,)Tj
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(and)Tj
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(5000)Tj
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2.285 0 Td
[(bits)-285(to)]TJ
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[(observ)15(e)-264(the)-264(beha)20(viour)-265(of)-264(our)-264(algorithm)-264(in)-265(hi)1(gher)-265(precision.)]TJ
0 -1.2 TD
[(F)15(or)-269(se)25(v)15(eral)-269(orders)]TJ
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7.472 0 Td
(m)Tj
/T1_1 1 Tf
1.147 0 Td
[(of)-269(composition)-269(doubling)-269(at)-269(each)-269(step,)]TJ
-8.619 -1.2 Td
[(we)-310(seek)-310(to)-309(\002nd)-310(the)-310(smallest)-310(v)25(alue)-310(of)-309(the)-310(number)-310(of)-310(points)]TJ
/T1_5 1 Tf
0 -1.2 TD
(n)Tj
/T1_1 1 Tf
0.936 0 Td
[(for)-335(which)-336(the)-335(bound)]TJ
/T1_5 1 Tf
8.673 0 Td
(B)Tj
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6.9738 0 0 6.9738 162.0369 442.0233 Tm
(math)Tj
9.9626 0 0 9.9626 179.8231 443.5177 Tm
[(on)-335(the)-336(mathematical)-335(error)-335(is)]TJ
-12.153 -1.2 Td
[(smaller)-246(than)-246(the)-246(bound)]TJ
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9.372 0 Td
(B)Tj
/T1_1 1 Tf
6.9738 0 0 6.9738 159.6802 430.0681 Tm
(rounding)Tj
9.9626 0 0 9.9626 187.8122 431.5625 Tm
[(on)-246(the)-246(rounding)-246(errors)-246(\(see)]TJ
-12.954 -1.2 Td
[(Figure)-215(1\).)-298(This)-215(v)25(alue)-215(of)]TJ
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9.604 0 Td
(n)Tj
/T1_1 1 Tf
0.815 0 Td
[(is)-215(considered)-215(optimal)-214(in)-215(the)-215(sense)]TJ
-10.419 -1.2 Td
[(that)-284(increasing)-284(it)-284(will)-284(decrease)]TJ
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12.585 0 Td
(B)Tj
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6.9738 0 0 6.9738 191.6835 406.1578 Tm
(math)Tj
9.9626 0 0 9.9626 208.9599 407.6522 Tm
[(with)-284(no)-284(bene\002t)-284(in)-285(the)]TJ
-15.077 -1.2 Td
[(guaranteed)-374(accurac)15(y)-374(since)]TJ
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11.102 0 Td
(B)Tj
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6.9738 0 0 6.9738 176.9154 394.2026 Tm
(rounding)Tj
9.9626 0 0 9.9626 206.3228 395.697 Tm
[(will)-374(increase,)-405(and)-374(us-)]TJ
-14.812 -1.2 Td
[(ing)-259(a)-259(smaller)-258(v)25(alue)-259(of)]TJ
/T1_5 1 Tf
8.934 0 Td
(n)Tj
/T1_1 1 Tf
0.859 0 Td
[(means)-259(that)-259(we)-258(are)-259(using)-259(too)-259(high)-259(a)]TJ
-9.793 -1.2 Td
[(w)10(orking)-301(precision.)-462(F)15(or)-301(each)-300(set)-301(of)-301(parameters)-300(we)-301(gi)25(v)15(e)-301(the)]TJ
0 -1.2 TD
[(number)-197(of)-196(good)-197(bits)-197(predicted)-197(by)-196(the)-197(softw)10(are,)-207(and)-197(the)-197(num-)]TJ
0 -1.2 TD
[(ber)-239(of)-238(bits)-239(actually)-239(correct,)-240(as)-239(measured)-239(ag)5(ainst)-238(a)-239(v)25(alue)-239(that)]TJ
T*
[(is)-344(assumed)-344(to)-344(be)-344(accurate)-344(to)-344(a)-344(precision)-345(higher)-344(than)-344(what)]TJ
T*
[(we)-290(will)-290(require)-291(aft)1(erw)10(ards.)-431(This)-290(reference)-290(v)25(alue)-291(w)10(as)-290(com-)]TJ
T*
[(puted)-306(with)-306(a)-306(precision)]TJ
/T1_5 1 Tf
9.334 0 Td
(p)Tj
/T1_4 1 Tf
0.885 0 Td
[(=)-381(5200)]TJ
/T1_1 1 Tf
3.465 0 Td
[(bits)-306(using)-306(the)]TJ
/T1_4 1 Tf
5.752 0 Td
(911)Tj
/T1_1 1 Tf
(-points)Tj
-19.436 -1.2 Td
[(Gauss-Le)15(gendre)-271(quadrature)-271(composed)]TJ
/T1_4 1 Tf
15.682 0 Td
(8)Tj
/T1_1 1 Tf
0.771 0 Td
[(times.)-373(F)15(or)-270(this)-271(set)]TJ
-16.452 -1.2 Td
[(of)-250(parameters)-250(our)-250(algorithm)-250(gi)25(v)15(es)]TJ
/T1_5 1 Tf
8.543 -2.263 Td
(B)Tj
/T1_1 1 Tf
6.9738 0 0 6.9738 151.4173 264.0649 Tm
(math)Tj
/T1_6 1 Tf
9.9626 0 0 9.9626 175.8256 265.5593 Tm
(\024)Tj
/T1_4 1 Tf
1.778 0 Td
(2)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 198.5183 269.6728 Tm
(\000)Tj
/T1_8 1 Tf
[(55)-1(99)]TJ
/T1_5 1 Tf
9.9626 0 0 9.9626 132.6256 250.6153 Tm
(B)Tj
/T1_1 1 Tf
6.9738 0 0 6.9738 140.1824 249.1209 Tm
(rounding)Tj
/T1_6 1 Tf
9.9626 0 0 9.9626 175.8256 250.6153 Tm
(\024)Tj
/T1_4 1 Tf
1.778 0 Td
(2)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 198.5183 254.7288 Tm
(\000)Tj
/T1_8 1 Tf
[(55)-1(94)]TJ
/T1_5 1 Tf
9.9626 0 0 9.9626 145.4089 235.6713 Tm
(B)Tj
/T1_1 1 Tf
6.9738 0 0 6.9738 152.9656 234.1769 Tm
(total)Tj
/T1_6 1 Tf
9.9626 0 0 9.9626 175.8256 235.6713 Tm
(\024)Tj
/T1_4 1 Tf
1.778 0 Td
(2)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 198.5183 239.7848 Tm
(\000)Tj
/T1_8 1 Tf
[(55)-1(93)]TJ
/T1_1 1 Tf
9.9626 0 0 9.9626 58.7518 213.1301 Tm
[(and)-316(a)-316(v)25(alue)]TJ
/T1_5 1 Tf
4.977 0 Td
(v)Tj
/T1_6 1 Tf
0.921 0 Td
(\031)Tj
/T1_4 1 Tf
1.178 0 Td
(1)Tj
/T1_5 1 Tf
(:)Tj
/T1_4 1 Tf
(011)Tj
/T1_6 1 Tf
2.549 0 Td
(\001)Tj
/T1_4 1 Tf
0.549 0 Td
(2)Tj
/T1_7 1 Tf
6.9738 0 0 6.9738 165.0855 216.7455 Tm
(\000)Tj
/T1_8 1 Tf
[(42)-1(1)]TJ
/T1_1 1 Tf
9.9626 0 0 9.9626 186.8724 213.1301 Tm
[(in)-316(binary)65(,)-333(so)-316(the)-316(computed)]TJ
-12.86 -1.2 Td
[(v)25(alue)-203(is)-204(accurate)-203(to)-204(about)]TJ
/T1_4 1 Tf
10.156 0 Td
(5593)Tj
/T1_6 1 Tf
2.05 0 Td
(\000)Tj
/T1_4 1 Tf
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