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Approximation
a172a174a173a34a175a177a176a18a17
a19 a69a17a59a99a89a41a59a23a78a79a97a50a64a66a57a60a98a99a77a17a95a92a59a5a62a128a72a27a97a102a68a41a96a117a61a19a68a41a98a99a77a17a101a87a71a73a57a92a69a17a90a110a71a73a80a17a59a85a57a60a69a75a71a79a59a19a90a70a78a73a72a41a95a46a57a92a69a20a6a149a202a8a2a70a192a47a96a114a68a41a78a185a72a12a77a53a72a41a69a17a59a19a95a21a9a102a62a63a80a17a57a65a61a81a80a125a57a60a64
a96a32a72a74a78a107a78a73a59a19a98a99a68a54a89a70a59a23a103a45a96a114a78a73a68a41a98 a77a53a72a74a69a17a59a23a95a63a203a81a192a153a57a65a64a180a71a79a68a45a64a79a57a92a98a99a77a17a95a60a97 a78a73a59a19a77a17a95a65a72a41a61a19a59a5a71a73a80a17a59a102a57a60a69a70a71a73a59a19a90a70a78a73a72a41a95a128a112a25a97a45a71a73a80a17a59a102a57a60a69a70a71a73a59a19a90a70a78a73a72a41a69a53a103
a59a19a89a54a72a41a95a92a101a53a72a74a71a79a59a48a103a45a72a54a71a185a71a73a80a17a59a111a61a19a59a19a69a75a71a79a78a73a68a41a57a65a103a174a68a74a96a63a71a73a80a17a59a5a77a53a72a41a69a17a59a19a95a1a9a40a118 a19 a96a91a61a9a68a70a101a17a78a81a64a66a59a70a192a47a71a73a80a17a57a60a64a181a57a65a64a67a71a79a68a25a68a174a64a66a57a60a98a99a77a17a95a92a57a65a64a66a71a79a57a65a61a107a96a114a68a41a78
a77a53a72a41a69a17a59a19a95a65a64a15a62a63a80a17a57a60a61a81a80a50a57a92a69a75a71a73a59a19a78a81a72a41a61a177a71a23a22a9a64a66a71a79a78a73a68a41a69a53a90a41a95a60a97a3a22a67a62a63a57a206a71a73a80a179a71a73a80a17a59a99a77a53a72a74a69a17a59a23a95a143a203a81a118a25a24a17a68a70a78a149a71a73a80a17a59a23a64a79a59a99a77a53a72a74a69a53a59a19a95a65a64a149a62a63a80a53a57a60a61a81a80a174a72a74a78a73a59
a61a9a95a60a68a70a64a79a59a19a78a135a112a25a97a41a192a30a62a128a59a180a98a107a72a27a97a50a101a53a64a66a59a107a72a12a96a114a68a70a101a17a78a66a195a38a77a43a68a41a57a60a69a70a71a67a57a60a69a75a71a79a59a19a90a70a78a73a72a74a71a79a57a60a68a41a69a125a64a79a61a81a80a53a59a19a98a99a59a41a118a107a104a105a69a45a68a70a101a17a78a67a98a99a57a92a69a53a103a53a64a67a62a153a59a85a64a79a77a17a95a60a57a206a71
a71a79a80a53a59a5a77a53a72a41a69a17a59a19a95a128a101a17a77a146a57a60a69a75a71a79a68a36a96a114a68a70a101a17a78a185a59a48a116a75a101a53a72a74a95a128a64a66a101a17a112a53a77a53a72a74a69a17a59a23a95a60a64a180a72a74a69a40a103a45a62a63a78a79a57a92a71a79a59a85a71a79a80a17a59a12a57a92a69a75a71a73a59a19a90a41a78a81a72a74a95a94a68a54a89a41a59a19a78a122a71a79a80a17a59a12a59a19a69a75a71a79a57a60a78a73a59
a77a53a72a41a69a17a59a19a95a26a9 a72a41a64a99a71a73a80a17a59a36a64a66a101a53a98 a68a74a96a15a96a114a68a41a101a53a78a107a57a92a69a75a71a79a59a23a90a41a78a81a72a74a95a65a64a99a68a41a69a21a71a79a80a17a59a48a64a66a59a110a96a114a68a70a101a17a78a5a64a79a101a17a112a17a77a53a72a41a69a17a59a19a95a65a64a23a118a134a191a117a80a17a59a19a69a157a101a53a64a79a59a110a71a79a80a17a59
a64a73a72a74a98a99a59a117a71a79a78a73a57a60a61a81a197a99a72a41a64a58a112a40a59a19a96a114a68a41a78a73a59a41a192a41a78a73a59a19a77a53a95a60a72a70a61a9a57a60a69a17a90a67a59a23a72a70a61a81a80a180a68a41a96a52a71a79a80a17a59a48a64a66a59a63a96a114a68a70a101a17a78a94a57a60a69a75a71a79a59a23a90a41a78a81a72a74a95a65a64a143a112a25a97a122a71a79a80a53a59a91a77a17a78a73a68a87a103a87a101a53a61a177a71a153a68a74a96a52a71a79a80a17a59
a57a60a69a70a71a73a59a19a90a70a78a73a72a41a69a53a103a34a192a17a59a23a89a27a72a41a95a92a101a40a72a54a71a79a59a48a103a110a72a54a71a63a71a79a80a53a59a181a61a9a59a23a69a75a71a79a78a73a68a41a57a65a103a5a68a41a96a46a59a48a72a41a61a81a80a110a68a74a96a46a71a73a80a17a59a23a64a79a59a122a64a66a101a53a112a17a77a53a72a74a69a53a59a19a95a65a64a76a72a74a69a53a103a12a71a73a80a17a59a185a72a41a78a79a59a48a72a181a68a41a96
a71a79a80a53a59a99a64a79a101a17a112a17a77a40a72a74a69a17a59a23a95a51a118a25a24a53a78a79a68a70a98 a57a92a69a75a71a73a101a87a71a79a57a60a68a41a69a50a62a153a59a180a112a40a59a23a95a92a57a60a59a19a89a70a59a180a71a79a80a53a72a74a71a149a71a73a80a17a57a60a64a67a64a73a61a81a80a17a59a23a98a180a59a180a57a65a64a149a90a70a68a41a57a60a69a17a90a85a71a73a68a110a90a41a57a60a89a41a59a185a101a40a64a15a72
a98a99a68a41a78a73a59a185a72a41a61a23a61a9a101a17a78a81a72a54a71a73a59a181a72a41a69a53a64a66a62a128a59a19a78a48a118a149a191a117a80a53a59a181a116a75a101a17a59a23a64a66a71a79a57a60a68a41a69a47a192a43a80a53a68a54a62a153a59a23a89a41a59a19a78a48a192a53a57a65a64a15a62a63a80a17a59a9a71a73a80a17a59a19a78a149a71a79a80a53a57a60a64a15a57a65a64a76a71a73a80a17a59a181a112a43a59a23a64a66a71a135a62a128a72a27a97
a71a79a68a67a90a70a68a67a68a41a78a94a72a41a78a79a59a128a71a73a80a17a59a19a78a73a59a76a112a43a59a9a71a66a71a73a59a19a78a94a71a79a59a23a61a81a80a53a69a17a57a60a116a75a101a17a59a48a64a28a27 a88 a72a41a69a180a62a128a59a76a103a87a68a67a71a73a80a17a59a91a64a73a72a74a98a99a59a63a197a25a57a92a69a40a103a99a68a74a96a52a57a92a69a75a71a79a59a23a90a41a78a81a72a54a71a73a57a92a68a70a69a181a68a70a69
a77a53a72a41a69a17a59a19a95a7a203a29a27a125a191a7a68a85a112a43a59a181a72a41a112a17a95a92a59a122a71a79a68a5a72a41a69a53a64a66a62a128a59a19a78a76a71a79a80a17a59a48a64a66a59a185a116a75a101a17a59a48a64a113a71a73a57a92a68a70a69a53a64a63a62a128a59a122a62a63a57a60a95a92a95a7a71a73a72a41a197a41a59a122a72a107a112a17a78a73a57a60a59a9a96a143a95a92a68a25a68a70a197a12a72a74a71a91a80a17a68a54a62
a69a25a101a17a98a99a59a19a78a73a57a60a61a23a72a74a95a52a57a60a69a70a71a73a59a19a90a70a78a73a72a74a71a79a57a60a68a41a69a12a57a60a64a117a77a43a59a19a78a79a96a114a68a41a78a73a98a99a59a23a103a34a192a17a72a181a208a53a59a23a95a60a103a12a68a74a96a143a64a66a71a79a101a53a103a87a97a5a197a75a69a53a68a54a62a63a69a12a72a70a64a30a22a9a116a75a101a53a72a41a103a17a78a73a72a74a71a79a101a17a78a73a59a31a22a74a118
a32
a120 a0 a136a2a1a4a3 a124a122a6a9a8a6a5a128a11a8a7 a0a76a121 a9a10a1a30a136a2a11a110a6a9a11a12a3
a13a117a138a66a139 a140a125a141a143a142a25a144a66a142a146a145a76a147a149a148a91a150a43a151a17a144a113a152a153a148a155a154a157a156a149a156a91a158a17a152a94a141a46a150a40a159
a14a47a161a32a163a43a161a32a163 a164 a37a43a55a32a55a32a37a47a24a41a14a87a20a48a29a32a37a43a28a16a15a12a29a32a16a48a24a74a39a54a22a75a20a23a29a18a17a41a14a87a20a48a29a32a37a43a28a84a37a40a35 a163 a15a184a13a149a31a34a33a30a14a87a20a48a29a32a37a43a28
a165a43a166a75a167a169a168a47a170a20a19
a193a22a21a32a207a24a23a26a25a28a27a16a29
a30a32a31
a21a114a207a34a33a66a207a24a35a36a23a66a196a4a21a114a207a24a35a37a23a39a38a41a40a4a35 a207a43a42a20a44a45a46a33a48a47a6a49
a88 a59a19a69a75a71a73a78a79a68a70a57a60a103a111a61a9a68a41a95a60a95a60a68a25a61a23a72a54a71a73a59a23a103a111a77a17a57a92a59a48a61a9a59a23a62a63a57a60a64a79a59a67a61a9a68a41a69a40a64a113a71a81a72a74a69a75a71a76a64a79a61a81a80a17a59a23a98a99a59
Normalized 1-D
Problem
Basis Function Approach
Collocation Discretization of
1-D Equation
() ( ) ( )
1
0
,xgxx x dSσ′′′Ψ=" [ ]0,1x ∈
0 0x = 1nx =1x 1nx ?2x
1σ nσ
1c
x
2c
x
nc
x
() ()
11
,
to be evaluated
j
i
jx
j
x
n
c
j
g xx dxSσ
?=
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!""#""$
a193a22a21a114a207a24a50a52a51a53a23a26a25a55a54a57a56
a2a53a58
a29
a196 a2a59a27a61a60a48a62
a60a62a39a63a65a64
a31
a21a32a207a24a50a52a51a66a33a79a207 a35a23a39a38a41a40 a35
a67 a68a6a69 a70
a204a72a71a8a73a52a74a75a74a77a76a6a78a66a79a81a80a82a78a81a204a72a74a84a83
a172a174a173a34a175a177a176a86a85
a187a162a59a19a71a73a64a117a71a81a72a74a197a70a59a185a72a107a64a66a57a60a98a99a77a17a95a60a59a122a59a9a209a17a72a74a98a99a77a17a95a60a59a67a57a60a69 a210 a190a181a118a53a191a117a80a17a59a185a103a17a68a41a98a107a72a74a57a60a69a102a57a60a64a76a71a79a80a17a59a181a64a66a59a23a90a41a98a99a59a19a69a75a71a87a44a45a46a33a88a47a88a49a47a68a74a96a46a71a73a80a17a59a122a78a73a59a23a72a74a95
a95a60a57a92a69a17a59a70a118a7a186a174a59a63a62a117a72a74a69a75a71a7a71a73a68a122a64a66a68a70a95a92a89a70a59a94a71a73a80a17a59a63a57a60a69a70a71a73a59a19a90a70a78a73a72a41a95a75a59a48a116a75a101a53a72a54a71a73a57a92a68a70a69a181a64a79a80a17a68a54a62a63a69a180a72a74a71a46a71a79a80a17a59a117a71a73a68a41a77a180a68a74a96a43a71a79a80a17a59a63a64a79a95a92a57a65a103a87a59a63a68a41a69a181a71a79a80a53a57a60a64
a103a87a68a70a98a99a72a41a57a92a69a47a118a117a191a117a80a53a59 a4a15a78a79a59a23a59a19a69a162a188 a64a117a96a114a101a17a69a40a61a177a71a79a57a60a68a41a69a179a57a60a64a149a103a87a59a19a69a17a68a41a71a79a59a48a103a110a112a75a97
a31
a21a114a207a89a33a79a207 a35a23a177a118a117a104a105a69a36a72a85a61a19a59a19a69a75a71a79a78a73a68a41a57a65a103a110a61a9a68a70a95a92a95a60a68a87a61a19a72a54a71a73a57a92a68a70a69
a64a73a61a81a80a17a59a19a98a99a59a41a192a30a62a128a59a85a103a87a57a92a89a25a57a65a103a87a59a107a71a73a80a17a59a5a103a87a68a70a98a99a72a41a57a92a69a125a57a60a69a75a71a79a68a36a198a84a64a79a59a19a90a41a98a99a59a23a69a70a71a81a64a90a44a207 a2 a33a66a207 a2a53a91
a29
a49 a9a92a25a93a45a46a33a88a94a48a94a88a94a6a33a79a198a143a192a46a62a63a57a206a71a73a80
a207
a30
a25a32a45a185a72a41a69a53a103a107a207
a56a96a95
a210 a118a46a191a117a80a17a59a149a61a81a80a40a72a74a78a73a90a41a59a76a103a87a59a23a69a53a64a66a57a92a71a113a97a107a196a102a57a65a64a153a72a70a64a79a64a79a101a17a98a99a59a23a103a99a71a73a68a185a112a43a59a91a77a17a57a60a59a23a61a19a59a19a62a63a57a65a64a66a59a15a61a9a68a70a69a53a64a113a71a81a72a74a69a75a71a94a68a70a69
a59a23a72a70a61a81a80a5a68a41a96a7a71a73a80a17a59a23a64a79a59a135a57a92a69a75a71a79a59a23a78a79a89a54a72a41a95a60a64a23a118a143a191a117a80a17a59a135a77a40a68a41a71a79a59a19a69a75a71a73a57a60a72a41a95a51a192a40a193 a57a60a64a117a71a73a80a17a59a19a69a12a59a19a89a54a72a41a95a92a101a53a72a74a71a79a59a48a103a111a72a74a71a128a71a73a80a17a59a122a61a9a59a23a69a75a71a79a78a73a68a41a57a65a103a17a64a128a207a24a50a52a51a177a118
a191a117a80a17a57a65a64a30a78a79a59a48a64a66a101a17a95a92a71a73a64a30a57a60a69a181a198a5a59a23a116a75a101a53a72a54a71a73a57a92a68a70a69a53a64a30a57a92a69a180a198a85a89a54a72a74a78a73a57a65a72a74a112a17a95a60a59a23a64a23a192a27a71a79a80a17a59a117a61a9a68a70a95a92a95a60a68a87a61a19a72a74a71a79a57a60a68a41a69a185a62a128a59a19a57a60a90a41a80a75a71a81a64 a0 a202 a33a79a203a59a25a97a47a98a33a88a94a48a94a88a94a6a33a79a198
a62a63a80a17a57a65a61a81a80a50a61a23a72a74a69a179a112a43a59a99a62a63a78a79a57a92a71a66a71a73a59a19a69a174a57a92a69a174a98a107a72a54a71a73a78a79a57a92a209a102a96a114a68a41a78a73a98a110a118 a19 a101a17a78a149a71a73a72a70a64a66a197a102a57a65a64a15a71a79a68a111a208a53a78a81a64a113a71a67a59a19a89a54a72a41a95a92a101a53a72a74a71a79a59a181a71a79a80a17a59a99a59a23a69a70a71a73a78a79a57a60a59a23a64
a68a74a96a153a71a79a80a17a59a107a98a107a72a74a71a79a78a73a57a206a209a179a72a41a69a53a103a50a64a79a101a17a112a53a64a79a59a23a116a75a101a17a59a23a69a75a71a79a95a60a97a179a64a66a68a70a95a92a89a70a59a185a71a73a80a17a59a85a64a79a59a9a71a67a68a74a96a128a59a23a116a75a101a53a72a54a71a73a57a92a68a70a69a53a64a23a118a22a99a91a68a74a71a79a59a180a71a73a80a53a72a54a71a67a71a73a80a17a59a99a203 a9 a204a32a205
a59a19a69a75a71a73a78a79a97 a68a74a96a67a71a73a80a17a59a179a98a107a72a54a71a73a78a79a57a92a209a84a57a60a64a12a72a74a69a84a57a60a69a75a71a79a59a23a90a41a78a81a72a74a95a91a68a41a96a67a71a79a80a17a59 a4a15a78a79a59a23a59a19a69a162a188 a64a5a96a114a101a17a69a53a61a177a71a73a57a92a68a70a69a162a192a76a59a23a89a27a72a41a95a92a101a40a72a54a71a79a59a48a103a134a72a54a71a111a71a79a80a17a59
a61a9a68a70a95a92a95a60a68a87a61a19a72a74a71a79a57a60a68a41a69a84a77a43a68a41a57a60a69a70a71a102a207 a50a51a81a192a149a68a54a89a70a59a19a78a85a71a73a80a17a59a174a57a92a69a75a71a79a59a23a78a79a89a54a72a41a95a100a44a207 a2a98a33a79a207 a2a53a91
a29
a49a38a192a91a62a63a80a17a57a65a61a81a80a199a57a65a64a111a71a79a80a17a59a174a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95a135a68a54a89a41a59a23a78
a62a63a80a17a57a65a61a81a80a102a71a79a80a53a59a185a112a53a72a70a64a66a57a65a64a63a96a114a101a17a69a40a61a177a71a79a57a60a68a41a69a174a201 a2a185a57a60a64a149a69a17a68a41a69a53a183a19a59a19a78a73a68a43a101a114a78a73a59a23a61a23a72a74a95a60a95a162a71a73a80a53a72a54a71a135a62a153a59a185a80a53a72a27a89a41a59a185a61a81a80a53a68a70a64a79a59a19a69a36a72a85a77a53a57a92a59a48a61a9a59a19a62a63a57a65a64a79a59
a61a9a68a70a69a53a64a66a71a73a72a74a69a75a71a185a72a41a77a17a77a17a78a73a68a27a209a25a57a60a98a107a72a54a71a73a57a92a68a70a69a96a102a177a118a111a104a38a96a113a192a7a80a17a68a54a62a128a59a19a89a70a59a19a78a48a192a47a62a128a59a85a103a87a59a23a61a19a57a60a103a17a59a23a103a174a71a79a68a50a61a81a80a17a68a25a68a70a64a79a59a99a72a102a103a17a57a206a182a52a59a19a78a73a59a19a69a75a71a181a64a79a59a9a71a185a68a41a96
a112a53a72a70a64a66a57a65a64a153a96a114a101a17a69a53a61a177a71a73a57a92a68a70a69a53a64a23a192a25a71a79a80a17a57a65a64a117a57a92a69a75a71a73a59a19a90a41a78a81a72a74a95a52a62a128a68a41a101a17a95a65a103a85a112a43a59a135a69a17a68a41a69a53a183a19a59a19a78a73a68a185a68a70a69a17a95a60a97a85a68a41a69a5a71a79a80a17a59a67a64a79a101a17a77a17a77a43a68a41a78a79a71a117a68a41a96a47a71a73a80a17a59a135a112a53a72a41a64a79a57a60a64
a96a114a101a17a69a53a61a9a71a79a57a60a68a41a69a53a64a103a101a32a68a41a78a48a192a63a71a79a68 a112a40a59a174a77a17a78a73a59a23a61a19a57a60a64a79a59a41a192a76a68a70a69a84a71a79a80a53a59a50a57a60a69a75a71a79a59a19a78a81a64a79a59a23a61a177a71a73a57a92a68a70a69a134a68a74a96a185a71a79a80a17a59a125a64a66a101a53a77a17a77a40a68a70a78a66a71a12a68a74a96a122a71a73a80a17a59a174a112a53a72a41a64a79a57a60a64
a96a114a101a17a69a53a61a9a71a79a57a60a68a41a69a102a72a74a69a40a103a92a44a45a46a33a48a47a6a49a37a102a177a118
a189
a13a117a138a51a137 a0a117a144a2a1 a156a4a3a6a5a8a7 a147a76a141a10a9a76a158a17a141a30a151a17a147a76a158a11a5a12a0a63a150a53a159a13a5a14a1a15a5
a165a43a166a75a167a169a168a47a170a17a16
a27 a29
a30a19a18
a21a114a207 a23a39a38a70a207a21a20
a18
a22
a47
a23a25a24
Normalized 1-D
Problem Simple Quadrature Scheme
()fx
x0 11
2
Area under the
curve is
approximated by
a rectangle
a172a174a173a34a175a177a176a21a26
a24a17a68a70a78a58a71a73a80a17a59a76a71a73a57a92a98a99a59a15a112a40a59a23a57a92a69a17a90a181a95a92a59a19a71a73a64a128a61a9a68a70a69a53a61a9a59a23a69a70a71a73a78a73a72a74a71a79a59a76a68a70a69a99a71a79a80a17a59a149a71a79a68a70a77a17a57a65a61a63a68a74a96a47a103a87a59a23a89a41a59a23a95a92a68a70a77a17a57a92a69a53a90a185a72a122a90a70a68a75a68a87a103a99a69a25a101a17a98a99a59a19a78a73a57a65a61a19a72a74a95
a71a79a59a48a61a81a80a17a69a17a57a65a116a75a101a17a59a76a96a114a68a70a78a153a59a23a89a27a72a41a95a92a101a40a72a54a71a79a57a60a69a17a90a67a71a73a80a17a59a15a57a92a69a75a71a79a59a23a90a41a78a81a72a74a95a40a68a74a96a47a72a67a96a114a101a53a69a53a61a177a71a73a57a92a68a70a69
a18
a21a114a207 a23a94a68a70a69a107a71a79a80a53a59a15a103a87a68a70a98a107a72a74a57a60a69a43a44a45 a33a88a47a6a49a38a118a46a186a50a59
a72a41a64a73a64a79a101a17a98a99a59a149a71a73a80a53a72a54a71a91a71a79a80a17a59a67a57a60a69a75a71a79a59a19a90a70a78a73a72a41a69a53a103a111a57a60a64a76a72a28a27a79a64a79a98a180a68a25a68a41a71a79a80a30a29a15a96a114a101a17a69a53a61a177a71a73a57a92a68a70a69a162a192a17a71a73a80a17a68a41a101a53a90a41a80a110a62a153a59a67a62a63a57a60a95a60a95a162a59a9a209a17a72a74a98a99a57a60a69a17a59a135a71a79a80a53a57a60a64
a72a41a64a73a64a79a101a17a98a99a77a87a71a79a57a60a68a41a69a12a95a60a72a74a71a79a59a23a78a23a118a1a24a30a57a92a78a81a64a66a71a76a62a153a59a67a72a74a78a73a59a135a90a41a68a70a57a92a69a17a90a181a71a79a68a85a103a17a59a19a89a41a59a23a95a92a68a70a77a12a72a180a69a53a72a41a57a92a89a70a59a67a72a74a77a17a77a17a78a73a68a70a72a70a61a81a80a85a96a114a68a70a78a63a68a41a112a87a71a81a72a74a57a60a69a17a57a92a69a53a90
a72a181a90a41a68a25a68a87a103a85a72a74a77a17a77a53a78a79a68a27a209a87a57a60a98a99a72a74a71a79a57a60a68a41a69a85a68a41a96a162a71a79a80a53a59a15a57a60a69a75a71a79a59a23a90a41a78a81a72a74a95a43a68a74a96a47a71a73a80a17a57a65a64a94a96a114a101a53a69a53a61a177a71a73a57a92a68a70a69a111a68a70a69a85a71a79a80a53a57a60a64a117a57a60a69a75a71a79a59a19a78a73a89a54a72a74a95a200a192a25a62a63a80a17a57a60a61a81a80a111a62a153a59
a61a19a72a41a95a92a95a52a71a73a80a17a59 a22a9a64a79a57a60a98a180a77a53a95a92a59a67a116a75a101a53a72a41a103a17a78a73a72a74a71a79a101a17a78a73a59a15a64a73a61a81a80a17a59a23a98a180a59 a22a54a118
a191a117a80a17a59a5a64a79a57a60a98a180a77a53a95a92a59a48a64a113a71a122a71a73a80a17a57a60a69a17a90a179a62a128a59a5a61a19a72a41a69a45a103a87a68a36a57a65a64a67a71a79a68a50a78a79a59a23a77a17a95a65a72a41a61a9a59a99a71a73a80a17a59a85a57a60a69a75a71a79a59a19a90a70a78a73a72a41a95a58a62a63a57a92a71a79a80a45a71a73a80a17a59a85a71a73a80a17a59a85a77a17a78a73a68a87a103a87a101a53a61a177a71
a68a74a96a30a71a79a80a53a59a67a57a92a69a75a71a79a59a23a90a41a78a81a72a74a69a53a103a162a192a87a59a19a89a54a72a74a95a60a101a53a72a54a71a73a59a23a103a12a72a54a71a91a72a180a77a43a68a41a57a60a69a70a71a76a57a60a69a53a64a79a57a60a103a17a59a135a71a79a80a17a59a122a57a60a69a75a71a79a59a23a78a79a89a54a72a74a95a200a192a17a72a74a69a40a103a5a71a73a80a17a59a67a95a60a59a19a69a17a90a41a71a79a80a110a68a74a96a30a71a79a80a17a59
a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95a200a192a87a62a63a80a17a57a60a61a81a80a110a57a60a69a12a71a79a80a17a57a65a64a76a61a19a72a70a64a66a59a135a57a65a64a63a101a17a69a17a57a92a71a113a97a41a118a94a104a38a96a30a62a128a59a122a61a81a80a17a68a25a68a70a64a79a59a91a71a73a80a17a59a122a77a43a68a41a57a60a69a75a71a76a68a74a96a7a59a23a89a54a72a74a95a60a101a53a72a54a71a73a57a92a68a70a69a110a72a70a64a128a71a79a80a17a59
a61a9a59a23a69a75a71a79a78a73a68a41a57a65a103a181a68a41a96a52a71a79a80a17a59a76a57a60a69a75a71a79a59a23a78a79a89a54a72a74a95a200a192a41a57a200a118 a59a41a118a46a207 a25a57a45 a94a31a17a192a41a62a128a59a76a61a19a72a41a95a92a95a53a71a79a80a17a59a91a64a73a61a81a80a17a59a23a98a180a59a25a22a9a98a99a57a60a103a87a77a43a68a41a57a60a69a75a71a153a116a75a101a53a72a41a103a17a78a73a72a74a71a79a101a17a78a73a59a31a22a74a118
a109 a98a99a57a60a103a17a77a40a68a70a57a92a69a75a71a111a116a75a101a53a72a70a103a87a78a73a72a74a71a79a101a17a78a73a59a110a64a79a61a81a80a17a59a23a98a99a59a12a78a73a59a19a77a53a95a60a72a70a61a9a59a23a64a180a71a73a80a17a59a36a72a74a78a73a59a23a72a174a101a17a69a40a103a87a59a19a78a107a71a73a80a17a59a36a61a9a101a53a78a79a89a70a59
a18
a21a32a207a24a23a99a112a25a97 a72
a78a73a59a23a61a177a71a81a72a74a69a17a90a70a95a92a59a63a62a63a80a53a68a70a64a79a59a76a80a17a59a19a57a60a90a41a80a75a71a94a57a65a64a143a71a73a80a17a59a76a96a114a101a53a69a53a61a177a71a73a57a92a68a70a69
a18
a21a114a207 a23a143a59a19a89a54a72a41a95a92a101a53a72a74a71a79a59a48a103a107a72a54a71a153a207 a25a32a45a46a94a32a31a87a118a46a191a117a80a17a59a91a64a73a61a81a80a17a59a23a98a180a59a91a57a60a64
a59a9a209a17a72a70a61a177a71a58a62a63a80a17a59a23a69
a18
a21a114a207 a23a143a57a65a64a94a72a67a61a9a68a70a69a53a64a66a71a73a72a74a69a75a71a48a118a34a33a91a68a54a62a153a59a23a89a41a59a19a78a48a192a74a62a63a80a53a72a74a71a94a57a65a64a58a95a60a59a23a64a73a64a58a68a41a112a25a89a25a57a60a68a41a101a53a64a143a57a65a64a143a71a73a80a53a72a54a71a94a71a79a80a17a59a15a64a79a61a81a80a17a59a23a98a99a59
a57a65a64a149a59a19a209a17a72a41a61a177a71a135a62a63a80a17a59a23a69
a18
a21a114a207 a23a149a57a60a64a135a72a111a95a92a57a60a69a17a59a23a72a41a78a91a96a114a101a53a69a53a61a177a71a73a57a92a68a70a69a50a68a41a96a94a207a45a72a41a64a149a62a153a59a23a95a92a95a200a118a185a191a117a80a17a59a180a98a99a68a70a64a66a71a15a68a70a112a75a89a25a57a60a68a41a101a53a64a149a62a117a72a27a97a12a68a41a96
a64a79a59a19a59a19a57a60a69a17a90a180a71a79a80a53a57a60a64a63a57a65a64a63a112a25a97a85a78a73a59a23a72a41a95a92a57a60a183a19a57a60a69a17a90a180a71a79a80a53a72a74a71a76a62a63a80a17a59a19a69
a18
a21a32a207a24a23a128a57a65a64a76a72a99a64a113a71a73a78a73a72a41a57a92a90a70a80a70a71a63a95a60a57a92a69a53a59a41a192a87a71a73a80a17a59a122a72a74a78a73a59a23a72a181a101a17a69a53a103a87a59a23a78a76a57a92a71a76a57a60a64
a72a111a71a79a78a81a72a74a77a43a59a19a183a23a68a41a57a65a103a34a118a180a191a117a80a53a57a60a64a15a71a73a78a73a72a41a77a40a59a23a183a19a68a41a57a65a103a36a80a40a72a41a64a135a59a9a209a17a72a70a61a177a71a79a95a60a97a110a71a79a80a53a59a85a64a79a72a41a98a180a59a99a72a74a78a73a59a23a72a12a72a41a64a15a71a73a80a17a59a99a78a79a59a48a61a177a71a81a72a74a69a17a90a70a95a92a59a180a62a63a80a17a57a65a61a81a80
a71a79a80a53a57a60a64a76a64a73a61a81a80a17a59a23a98a180a59a135a101a53a64a79a59a23a64a128a71a73a68a107a72a74a77a17a77a17a78a73a68a27a209a87a57a92a98a107a72a74a71a79a59a149a71a79a80a17a59a67a57a60a69a75a71a79a59a19a90a70a78a73a72a41a95a4a101a32a61a19a72a41a69a110a97a41a68a41a101a12a64a66a59a23a59a67a62a63a80a25a97 a27 a102a9a118
a187a162a59a19a71a73a64a76a71a79a78a73a97a111a71a73a68a5a103a87a59a23a78a79a57a60a89a41a59a67a71a73a80a17a57a65a64a91a57a60a69a179a72a85a64a79a95a92a57a60a90a41a80a75a71a73a95a92a97a110a103a87a57a92a182a43a59a23a78a79a59a23a69a70a71a15a62a128a72a27a97a70a118a63a104a105a69a53a64a66a71a79a59a23a72a70a103a110a68a74a96a143a71a73a80a17a59a185a57a60a69a75a71a79a59a19a78a73a89a54a72a74a95a7a112a40a59a23a57a92a69a17a90
a44a45a46a33a88a47a88a49a153a62a128a59a111a61a19a68a41a69a53a64a79a57a60a103a17a59a19a78a181a72a41a69a146a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95 a44a45a46a33a36a35 a49a52a33a37a35a39a38 a45a102a71a73a68a179a112a43a59a111a72a179a112a17a57a206a71a99a98a99a68a41a78a73a59a85a90a41a59a23a69a17a59a19a78a81a72a74a95a200a118a36a186a174a59a111a98a107a72a27a97
a59a9a209a17a72a41a77a17a69a53a103
a18
a21a114a207a24a23a117a72a41a112a40a68a70a101a87a71a63a71a79a80a53a59a122a61a9a59a19a69a75a71a73a78a79a68a70a57a60a103a5a68a74a96a30a71a79a80a17a57a65a64a63a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95a200a192a41a40a207 a25 a205a42
a18
a21a114a207 a23a26a25
a18
a21a43a40a207a24a23a14a44a46a45 a21a114a207 a23
a38
a18
a21a47a40a207 a23
a38a70a207
a44
a45 a21a32a207a24a23
a42
a23a49a48
a38
a42
a18
a21a51a50a98a23
a38a70a207
a42
a18a53a52a55a54a57a56a58a52a55a59a61a60
a50a22a42a103a44a45a46a33a36a35 a49
a62a63a80a17a59a23a78a79a59a62a45 a21a32a207a24a23a61a25 a207a64a63a65a40a207a7a118a132a191a117a80a53a59a50a95a65a72a41a64a66a71a111a71a73a59a19a78a73a98 a57a92a69 a71a79a80a53a59a174a59a19a209a87a77a53a72a74a69a53a64a79a57a60a68a41a69a134a57a65a64a111a71a79a80a17a59a45a191a30a72a27a97a75a95a60a68a41a78a12a64a66a59a23a78a79a57a60a59a23a64
a78a73a59a19a98a107a72a74a57a60a69a53a103a87a59a23a78a23a118a46a187a162a59a19a71a73a64a63a57a60a69a75a71a79a59a19a90a70a78a73a72a74a71a79a59a149a71a79a80a17a57a65a64a63a59a19a209a87a72a41a77a53a72a74a69a40a64a66a57a60a68a41a69a111a68a54a89a41a59a19a78a128a71a73a80a17a59a67a57a92a69a75a71a73a59a19a78a73a89a27a72a41a95a26a44a45 a33a37a35a46a49
a27
a205
a30 a18
a21a114a207 a23a39a38a70a207 a25a66a35
a18
a21a43a40a207a24a23a14a44
a35a11a67
a23a69a68
a38
a42
a18
a21a70a50a98a23
a38a70a207
a42
a33a76a59a23a69a53a61a9a59a15a71a73a80a17a59a67a59a19a78a73a78a79a68a70a78a117a57a92a69a12a71a79a80a53a59a67a98a180a57a65a103a87a77a43a68a41a57a60a69a75a71a91a116a70a101a40a72a41a103a87a78a81a72a54a71a73a101a17a78a79a59a135a72a74a77a53a77a17a78a79a68a27a209a87a57a60a98a107a72a54a71a79a57a60a68a41a69a5a57a60a64
a71 a25 a27
a205
a30 a18
a21a114a207a24a23a84a38a41a207a72a63a62a35
a18
a21a47a40a207 a23 a25
a35a73a67
a23a69a68
a38
a42
a18
a21a70a50a98a23
a38a70a207
a42
a74
a109 a64a66a71a79a78a81a72a74a57a60a90a41a80a75a71a58a95a60a57a92a69a53a59a91a68a41a78a128a72a122a61a9a68a70a69a53a64a66a71a73a72a74a69a75a71a94a62a128a68a41a101a53a95a60a103a99a90a70a59a19a69a17a59a23a78a73a72a74a71a79a59a76a72a67a183a23a59a19a78a73a68a185a72a41a64a58a71a79a80a17a59a15a64a66a59a48a61a9a68a41a69a40a103a107a103a87a59a19a78a73a57a92a89a54a72a74a71a79a57a60a89a41a59a76a72a74a69a40a103
a71a79a80a53a59a19a78a73a59a9a96a114a68a41a78a73a59a63a71a73a80a17a59a15a72a41a112a40a68a54a89a70a59a76a59a9a209a87a77a17a78a73a59a23a64a73a64a79a57a92a68a70a69a99a71a79a59a19a95a60a95a65a64a153a101a40a64a58a71a73a80a53a72a54a71a128a71a79a80a17a59a15a59a19a78a73a78a73a68a41a78a94a57a65a64a153a57a60a103a87a59a23a69a75a71a79a57a65a61a19a72a74a95a60a95a60a97a180a183a23a59a19a78a73a68a122a96a114a68a70a78a153a112a43a68a74a71a73a80
a71a79a80a53a59a23a64a79a59a135a61a23a72a41a64a79a59a23a64a23a118
a99a76a68a54a62a67a192a117a72a74a69a53a68a74a71a79a80a53a59a19a78a99a57a92a98a99a77a43a68a41a78a79a71a73a72a41a69a70a71a85a77a53a57a92a59a48a61a9a59a110a68a74a96a135a57a92a69a17a96a114a68a41a78a73a98a99a72a74a71a79a57a60a68a41a69a21a71a79a80a53a72a74a71a85a62a153a59a102a68a70a112a87a71a73a72a41a57a92a69a21a96a114a78a73a68a41a98 a71a73a80a17a57a60a64a85a95a92a57a92a71a66a71a73a95a92a59
a59a9a209a87a59a23a78a73a61a19a57a60a64a79a59a107a57a60a64a185a71a79a80a40a72a54a71a180a71a79a80a17a59a111a59a19a78a73a78a79a68a70a78a185a64a79a61a23a72a74a95a60a59a23a64a181a72a41a64a67a71a73a80a17a59a110a61a9a101a17a112a43a59a5a68a41a96a63a71a79a80a17a59a12a103a87a68a41a98a107a72a74a57a60a69a146a95a60a59a19a69a17a90a41a71a79a80a162a118a174a191a117a80a17a57a65a64a122a71a73a59a19a95a60a95a60a64
a101a53a64a76a71a79a80a53a72a74a71a76a96a114a68a41a78a91a64a79a98a107a72a74a95a60a95a47a59a23a69a17a68a41a101a17a90a70a80a110a57a92a69a75a71a79a59a23a78a79a89a54a72a41a95a60a64a63a68a70a101a17a78a149a64a79a57a60a98a180a77a53a95a92a59a122a78a73a101a17a95a60a59a67a57a60a64 a19a1a0 a112a53a101a87a71a91a96a114a68a70a78a76a95a60a72a41a78a79a90a70a59a19a78a91a103a17a68a41a98a107a72a74a57a60a69
a95a60a59a19a69a17a90a41a71a79a80a53a64a117a62a128a59a135a80a53a72a27a89a41a59a15a71a79a68a107a64a79a59a19a59a23a197a85a112a40a59a19a71a66a71a73a59a19a78a76a98a99a59a9a71a73a80a17a68a87a103a17a64a23a118
a14a47a161a65a160a162a161a32a163 a2 a42a146a44a7a39a54a37a4a3a162a29a32a28a30a49a174a20a23a26a30a22a6a5a99a24a41a24a70a33a47a39a54a14a17a24a41a18
a165a43a166a75a167a169a168a47a170a8a7
a27 a29
a30 a18
a21a114a207a24a23a84a38a41a207 a20
a47
a23
a18
a22 a47
a68 a24
a44
a47
a23
a18
a22a10a9
a68 a24
Normalized 1-D
Problem
Simple Quadrature Scheme
Area under the
curve is
approximated by
two rectangles
Improving the Accuracy
()fx
x0 11
4
3
4
1
2
a172a174a173a34a175a177a176a6a11
a19 a69a17a59a58a62a117a72a27a97a15a68a74a96a17a57a60a98a180a77a53a78a79a68a54a89a25a57a60a69a17a90a63a98a107a72a27a97a149a112a43a59a94a71a73a68a91a103a87a57a60a89a25a57a60a103a87a59a94a71a79a80a17a59a128a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95a96a44a45a46a33a48a47a6a49a75a57a92a69a75a71a79a68a149a64a66a101a17a112a53a57a92a69a75a71a79a59a23a78a79a89a54a72a41a95a60a64 a44a45 a33a53a45a46a94a32a31 a49
a72a74a69a40a103a92a44a45a46a94a32a31 a33a48a47a6a49a162a72a74a69a53a103a12a62a63a78a73a57a206a71a73a59a15a71a79a80a17a59a67a57a60a69a75a71a79a59a19a90a70a78a73a72a41a95
a27 a29
a30 a18
a21a32a207a24a23a39a38a70a207 a25 a27
a30a13a12a14
a30 a18
a21a32a207a24a23a39a38a70a207 a44 a27 a29
a30a13a12a14 a18
a21a114a207 a23a39a38a70a207
a72a74a69a40a103 a72a74a77a53a77a17a95a92a97a146a72a174a98a99a57a65a103a87a77a40a68a70a57a92a69a75a71a107a78a73a101a17a95a92a59a111a71a79a68a174a71a73a80a17a59a110a57a92a69a75a71a79a59a23a90a41a78a81a72a74a95a128a68a41a69a21a59a23a72a41a61a81a80 a64a66a101a53a112a17a57a92a69a75a71a73a59a19a78a73a89a27a72a41a95a51a118 a186a50a59a110a68a41a112a17a71a73a72a74a57a60a69a21a72
a64a73a61a81a80a17a59a19a98a99a59a99a64a66a80a17a68a54a62a63a69a174a57a60a69a179a71a73a80a17a59a85a64a66a95a60a57a65a103a87a59a41a118a107a191a117a80a17a59a180a96a32a72a41a61a9a71a79a68a70a78
a29
a42 a72a74a77a17a77a43a59a23a72a41a78a79a57a60a69a17a90a12a57a92a69a50a96a114a78a79a68a70a69a75a71a135a68a41a96
a18
a21
a29
a42 a23a135a72a41a69a53a103
a18
a21 a67
a15
a23
a72a74a78a73a59a17a16a113a101a53a64a113a71a63a71a73a80a17a59a122a103a87a68a70a98a99a72a41a57a92a69a110a95a60a59a19a69a17a90a41a71a79a80a53a64a23a118
a18a20a19a22a21
a176a24a23a26a25a28a27a30a29a41a176a146a178 a88 a72a41a69a179a97a70a68a41a101a179a61a19a68a41a98a99a59a185a101a17a77a50a62a63a57a206a71a73a80a174a72a41a69a36a59a9a209a87a77a17a78a73a59a23a64a73a64a79a57a92a68a70a69a110a96a114a68a41a78a15a71a79a80a17a59a180a59a19a78a73a78a73a68a41a78a91a57a60a69a36a71a73a80a17a57a65a64a15a61a23a72a41a64a79a59 a27
a33a76a68a54a62 a98a185a101a53a61a81a80a110a103a87a68a25a59a23a64a128a71a73a80a17a59a122a72a41a61a23a61a9a101a17a78a81a72a41a61a19a97a107a57a92a98a99a77a17a78a73a68a54a89a41a59 a27
a14a47a161a65a160a162a161a65a160 a82 a22a25a28a30a22a70a39a54a14a17a55a117a28a32a31a34a33a149a37a40a29a51a28a52a20a36a35a7a37a53a39a54a42a174a33a7a55a51a14 a165a43a166a75a167a169a168a47a170a8a37
a27 a29
a30 a18
a21a32a207a24a23a39a38a70a207a61a20
a56
a38
a202a58
a29
a47
a198
a18
a22 a203 a63
a29
a42
a198 a24
SMA-HPC ?1999 MIT
Normalized 1-D
Problem
Simple Quadrature Scheme
()f x
x0 1
a39
a0 a59a19a97a111a116a70a101a53a59a23a64a66a71a79a57a60a68a41a69a53a64a63a72a41a112a40a68a70a101a87a71a117a71a79a80a17a59a67a98a99a59a9a71a73a80a17a68a87a103a1a0
a2a102a37a4a3 a35a38a14a17a16a23a20a99a86a30a37a125a20a23a26a30a22a36a22a75a39a27a39a54a37a53a39a54a16a107a86a30a22a75a24a70a14a41a18a5a3a107a29a114a20a48a26 a28a7a6
a5a85a39a54a22a102a20a48a26a7a22a75a39a54a22a9a8a94a22a75a20a23a20a23a22a75a39a85a42a146a22a75a20a23a26a30a37a162a86a30a16a10a6
a172a174a173a34a175a177a176a6a11
a109 a64a91a97a41a68a70a101a110a61a23a72a74a69a36a64a79a59a19a59a185a103a17a57a92a89a25a57a65a103a87a57a92a69a53a90a107a71a79a80a17a59a185a57a92a69a75a71a79a59a23a78a79a89a54a72a41a95a47a57a60a69a70a71a73a68a99a71a113a62a153a68a5a78a79a59a48a103a87a101a53a61a9a59a48a64a117a71a79a80a53a59a185a59a19a78a73a78a73a68a41a78a76a72a74a69a40a103a12a71a79a80a17a59a23a78a79a59a67a57a65a64a76a69a17a68
a78a73a59a23a72a41a64a79a68a41a69a146a71a79a68a45a64a113a71a73a68a41a77a157a72a54a71 a16a113a101a53a64a66a71a107a71a113a62a153a68a125a64a79a101a17a112a17a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95a65a64a99a62a63a80a17a59a19a69 a62a153a59a102a61a23a72a74a69 a80a40a72a27a89a41a59a12a198 a64a79a101a17a112a17a57a60a69a75a71a79a59a19a89a54a72a41a95a60a64a99a72a74a69a40a103
a78a73a59a19a77a43a59a23a72a54a71a143a68a70a101a17a78a143a98a99a57a60a103a87a77a43a68a41a57a60a69a75a71a153a116a75a101a53a72a41a103a17a78a73a72a74a71a79a101a17a78a73a59a153a78a73a101a17a95a60a59a63a68a41a69a99a59a23a72a70a61a81a80a180a64a79a101a17a112a17a57a60a69a75a71a79a59a23a78a79a89a54a72a74a95a200a118a47a186a174a59a76a68a70a112a87a71a73a72a41a57a92a69a181a71a79a80a17a59a149a64a79a61a81a80a17a59a23a98a99a59
a27 a29
a30 a18
a21a114a207 a23a39a38a70a207 a25
a56
a38
a202a58
a29
a47
a198
a67a68a6a69a70
a64a79a101a17a112a17a57a60a69a75a71a79a59a19a78a73a89a54a72a74a95
a95a60a59a19a69a17a90a41a71a79a80
a18
a21a32a207 a50a51a39a23
a62a63a80a17a59a23a78a79a59a67a71a73a80a17a59a99a61a9a59a19a69a75a71a73a78a79a68a70a57a60a103a110a68a41a96a58a71a79a80a17a59a181a203a38a204a32a205a5a64a79a101a17a112a17a57a60a69a75a71a79a59a19a78a73a89a54a72a74a95a30a57a60a64a149a207 a50a51a8a25
a29
a42 a21
a202a12a11
a29
a56
a44
a202
a56
a23a12a25
a202a13a11
a64
a14
a56
a118a67a191a117a80a17a59a23a78a79a59a185a57a60a64
a69a17a68a110a103a87a68a41a101a53a112a87a71a67a71a79a80a53a72a74a71a67a62a153a59a99a90a70a72a41a57a92a69a47a192a34a112a17a101a87a71a67a71a79a80a53a59a180a197a70a59a19a97a179a116a75a101a17a59a48a64a113a71a73a57a92a68a70a69a179a57a65a64a135a112a25a97a36a80a17a68a54a62a194a98a181a101a53a61a81a80 a27 a33a76a68a54a62 a103a87a68a25a59a23a64a15a71a79a80a53a57a60a64
a90a70a72a41a57a92a69a50a64a79a61a23a72a74a95a60a59a181a62a63a57a92a71a79a80a50a71a79a80a17a59a180a69a25a101a17a98a185a112a43a59a19a78a135a68a41a96a153a64a66a101a53a112a17a57a92a69a75a71a73a59a19a78a73a89a27a72a41a95a60a64a15a101a53a64a79a59a23a103 a27 a109 a69a40a103a36a208a53a69a40a72a74a95a60a95a92a97a70a192a162a72a74a78a73a59a122a71a79a80a53a59a19a78a73a59a180a61a19a95a92a59a23a89a41a59a19a78
a62a117a72a27a97a25a64a128a68a41a96a7a68a41a112a17a71a73a72a74a57a60a69a17a57a60a69a17a90a99a112a40a59a19a71a66a71a73a59a19a78a91a72a70a61a19a61a9a101a53a78a73a72a70a61a9a97a107a62a63a57a92a71a79a80a110a95a92a59a48a64a79a64a117a59a19a182a43a68a70a78a66a71 a27
a14a47a161a65a160a162a161a14 a15 a33a30a42a146a22a75a39a54a29a32a24a41a14a53a55a94a13 a13a162a14a17a42a45a44a46a55a32a22
a165a43a166a75a167a169a168a47a170a17a16
a27 a29
a30a19a56
a203a200a198a26a21a32a207a24a23a84a38a41a207a21a20
a56
a38
a202a58
a29
a47
a198
a56
a203a38a198
a22 a203a14a63
a29
a42
a198 a24
SMA-HPC ?1999 MIT
Normalized 1-D
Problem
Simple Quadrature Scheme
Numerical Example
E
r
r
o
r
n
a172a174a173a34a175a177a176a19a18
a187a162a59a19a71a73a64a111a95a92a68a25a68a70a197a157a72a74a71a12a72a41a69a134a59a9a209a17a72a74a98a99a77a17a95a60a59a179a68a41a96a122a57a92a69a75a71a73a59a19a90a41a78a81a72a54a71a73a57a92a69a53a90
a18
a21a114a207 a23a92a25
a56
a203a200a198a26a21a32a207a24a23a5a68a41a69a134a68a70a101a17a78a110a103a87a68a41a98a107a72a74a57a60a69a162a118 a186a50a59
a68a41a112a17a71a73a72a74a57a60a69a12a77a17a78a79a68a70a90a41a78a73a59a23a64a73a64a66a57a60a89a41a59a23a95a92a97a180a112a43a59a9a71a66a71a73a59a19a78a76a72a41a69a53a64a66a62a128a59a19a78a81a64a153a71a79a68a180a71a79a80a53a59a67a57a92a69a75a71a79a59a23a90a41a78a81a72a74a95a52a112a25a97a85a57a60a69a53a61a9a78a73a59a23a72a70a64a66a57a60a69a17a90a181a71a79a80a17a59a135a69a25a101a17a98a185a112a43a59a19a78a63a68a41a96
a64a79a101a17a112a17a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95a65a64a91a198a143a118a67a191a117a80a17a59a180a59a19a78a73a78a79a68a70a78a91a57a60a69a179a59a23a89a27a72a41a95a92a101a40a72a54a71a79a57a60a69a17a90a5a71a79a80a17a59a99a57a60a69a70a71a73a59a19a90a70a78a73a72a41a95a47a57a65a64a149a77a53a95a92a68a41a71a66a71a79a59a48a103a179a72a70a64a149a72a5a96a114a101a17a69a53a61a9a71a79a57a60a68a41a69a179a68a41a96
a71a79a80a53a59a149a69a25a101a17a98a181a112a40a59a23a78a128a68a41a96a47a64a66a101a53a112a17a57a92a69a75a71a73a59a19a78a73a89a27a72a41a95a60a64 a101a114a198a89a102a9a118a58a191a117a80a17a59a15a59a19a78a73a78a79a68a70a78a153a72a74a77a53a77a40a59a48a72a74a78a81a64a58a71a73a68a181a112a43a59a149a90a70a68a41a57a60a69a17a90a181a103a17a68a54a62a63a69a5a72a41a64a21a20a23a22
a29
a56
a14a4a24
a118
a187a162a59a19a71a73a64a63a64a79a59a19a59a135a62a63a80a25a97a41a118
a24a17a78a73a68a41a98 a62a63a80a53a72a54a71a107a62a128a59a12a80a40a72a27a89a41a59 a16a113a101a53a64a113a71a5a64a79a59a19a59a23a69a162a192a153a71a79a80a17a59a102a59a23a78a79a78a73a68a41a78a180a57a92a69a40a64a66a57a65a103a87a59a110a71a79a80a53a59a12a203a38a204a32a205a45a64a79a101a17a112a17a57a60a69a75a71a79a59a23a78a79a89a54a72a74a95a2a101a32a68a74a96a135a95a92a59a23a69a17a90a74a71a73a80
a25 a35 a25
a29
a56
a188a102a63a57a65a64 a205a27a26a42
a15
a83
a14a29a28a31a30a33a32
a51a13a34
a83
a60
a14
a96a114a68a70a78a91a64a79a68a41a98a99a59a28a50a19a202 a42 a44
a202a13a11
a29
a56
a33
a202
a56
a49a200a118 a33a91a59a19a69a53a61a19a59a41a192a40a96a114a68a41a78a63a71a73a80a17a59a185a59a23a69a75a71a79a57a60a78a79a59a67a57a60a69a75a71a79a59a19a78a73a89a54a72a74a95 a44a45 a33a88a47a6a49
a62a128a59a180a61a23a72a74a69a125a64a66a101a17a98 a71a79a80a17a59a48a64a66a59a180a59a19a78a73a78a79a68a70a78a73a64a15a72a74a69a40a103a179a68a70a112a87a71a73a72a41a57a92a69a179a71a79a80a17a59a99a59a19a78a73a78a73a68a41a78a48a192a53a71
a56
a96a114a68a70a78a135a72a41a69a174a72a41a77a17a77a17a78a73a68a27a209a87a57a92a98a107a72a54a71a73a57a92a68a70a69a102a101a53a64a79a57a92a69a17a90
a35
a188 a198a143a188a17a64a66a101a17a112a53a57a92a69a75a71a79a59a23a78a79a89a54a72a41a95a60a64a63a72a70a64
a71
a56
a25
a198a14a35a73a67
a23a69a68
a0
a47
a198
a56
a38
a202a58
a29
a38
a42
a18
a21a70a50a23a202a18a23
a38a70a207
a42a2a1
a67 a68a6a69 a70
a50
a78a66a79a79a52a204a32a205
a202a4a3a6a5a7a9a8
a104a38a71a143a57a60a64a58a59a23a72a41a64a79a97a135a71a79a68a67a64a79a59a19a59a128a71a79a80a40a72a54a71a58a57a92a96
a18
a21a32a207a24a23a30a57a65a64a143a72a15a61a19a68a41a69a75a71a79a57a60a69a25a101a17a68a41a101a40a64a47a96a114a101a53a69a53a61a177a71a73a57a92a68a70a69a162a192 a25a11a10 a188a46a101a114a112a43a59a19a57a60a69a17a90a135a71a79a80a53a59a63a98a180a59a48a72a74a69a96a102a7a98a181a101a53a64a113a71
a112a43a59a107a112a40a68a70a101a17a69a53a103a87a59a48a103a179a112a25a97a102a71a73a80a17a59a107a98a107a72a54a209a87a57a92a98a181a101a17a98 a72a41a69a53a103a36a71a73a80a17a59a107a98a99a57a92a69a25a101a17a98a181a101a17a98 a68a41a96
a18
a21a32a207a24a23a15a68a41a69a50a71a79a80a53a59a107a57a92a69a75a71a79a59a23a78a79a89a54a72a41a95a12a44a45 a33a88a47a6a49
a72a74a69a40a103a110a80a17a59a19a69a53a61a19a59a41a192a43a71a79a80a53a59a19a78a73a59a122a98a185a101a53a64a66a71a149a59a19a209a87a57a60a64a66a71a15a64a79a68a41a98a99a59a28a50 a42a16a44a45a46a33a88a47a88a49a7a64a79a101a53a61a81a80a102a71a79a80a40a72a54a71 a10 a25a10a38
a42
a18
a21a70a50a98a23a13a12a82a38a70a207
a42
a118a13a33a76a59a23a69a53a61a9a59
a62a128a59a15a68a70a112a87a71a73a72a41a57a92a69a111a71a79a80a53a59a67a59a23a64a66a71a79a57a60a98a99a72a74a71a79a59
a71
a56
a25
a198a14a35a11a67
a23a69a68
a38
a42
a18
a21a70a50a98a23
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a61a9a68a70a98a99a59a23a64a67a96a114a78a73a68a41a98 a71a79a80a53a59a111a64a66a71a79a78a73a101a53a61a177a71a73a101a17a78a79a59a5a68a74a96a76a191a7a72a27a97a25a95a60a68a41a78a48a188 a64a185a64a66a59a23a78a79a57a60a59a23a64a67a59a19a209a87a77a53a72a74a69a53a64a79a57a60a68a41a69a162a118a36a191a117a80a17a59a111a191a7a72a27a97a25a95a60a68a41a78a122a59a19a209a25a77a40a72a74a69a53a64a79a57a92a68a70a69
a43
a68a74a96a58a72a99a96a114a101a53a69a53a61a177a71a73a57a92a68a70a69a36a57a92a69a36a72a85a95a92a68a87a61a19a72a41a95a47a69a17a59a23a57a92a90a70a80a25a112a40a68a70a78a79a80a17a68a25a68a87a103a12a68a74a96a94a72a99a77a43a68a41a57a60a69a75a71a22a101a114a80a53a59a19a78a73a59a67a71a79a80a17a57a65a64a149a77a40a68a70a57a92a69a75a71a91a57a65a64a91a61a81a80a17a68a75a64a66a59a23a69a36a72a41a64a1a0
a62a63a57a92a71a79a80a17a68a70a101a87a71a76a95a92a68a75a64a79a64a128a68a41a96a30a90a70a59a19a69a17a59a23a78a73a72a41a95a92a57a92a71a113a97a46a102a58a57a65a64a117a69a17a68a74a71a73a80a17a57a60a69a17a90a107a112a17a101a87a71a91a72a180a77a43a68a54a62a153a59a23a78a117a64a79a59a19a78a73a57a60a59a23a64a117a59a9a209a87a77a53a72a41a69a53a64a79a57a92a68a70a69 a15a30a191a117a80a53a59a67a80a17a57a92a90a70a80a17a59a19a78
a71a79a80a53a59a122a68a41a78a81a103a87a59a19a78a63a68a41a96a30a77a43a68a41a95a60a97a25a69a17a68a41a98a99a57a65a72a74a95a65a64a117a71a79a80a40a72a54a71a76a68a41a101a53a78a91a64a79a61a81a80a53a59a19a98a99a59a122a61a19a72a41a69a12a57a60a69a75a71a79a59a23a90a41a78a81a72a54a71a79a59a135a71a73a80a17a59a67a80a17a57a60a90a41a80a17a59a23a78a63a71a79a80a53a59a122a68a41a78a81a103a87a59a19a78a63a68a41a96
a71a79a80a53a59a111a78a73a59a19a98a107a72a74a57a60a69a53a103a87a59a23a78a180a71a79a59a19a78a73a98 a57a60a69 a71a73a80a17a59a111a59a19a209a87a77a53a72a74a69a53a64a79a57a60a68a41a69a162a118 a191a117a80a53a59a111a57a60a69a75a71a79a59a19a90a70a78a73a72a41a95a153a68a41a96a76a71a79a80a53a59a12a78a73a59a19a98a107a72a41a57a92a69a17a59a23a78a181a68a54a89a70a59a19a78a181a71a79a80a17a59
a103a87a68a70a98a99a72a41a57a92a69a12a57a65a64a117a77a17a78a79a59a48a61a9a57a65a64a66a59a23a95a92a97a85a71a79a80a17a59a67a59a23a78a79a78a73a68a41a78a128a57a92a69a110a69a25a101a17a98a99a59a19a78a73a57a65a61a19a72a74a95a34a57a60a69a75a71a79a59a23a90a41a78a81a72a54a71a79a57a60a68a41a69a47a118
a14a47a161a14a47a161a3a2 a13a15a16a19a20a48a29a32a42a146a14a17a20a23a29a32a28a30a49a50a20a23a26a30a22a102a13a91a39a27a39a54a37a53a39
a165a43a166a75a167a169a168a47a170a21a171a17a171
a4a91a64a79a57a60a69a17a90a180a71a79a80a17a59a122a191a30a72a27a97a25a95a92a68a70a78a117a64a79a59a19a78a73a57a60a59a23a64a128a78a79a59a48a64a66a101a53a95a206a71a81a64a63a72a74a69a53a103a111a71a79a80a53a59a67a59a9a209a17a72a41a61a9a71a79a69a17a59a48a64a79a64a117a61a9a78a73a57a92a71a79a59a19a78a73a57a65a72
a5
a29a30
a18
a21a32a207a24a23a39a38a70a207a21a63 a54a57a56
a202a58
a29
a20
a202
a18
a21a32a207a40a202a52a23 a25
a47
a21a38a18a11a44 a47a23 a48
a27a16a29
a30
a33
a79
a91
a29
a18
a21a12a39a207 a21a114a207a24a23a53a23
a33
a207 a79
a91
a29
a207 a79
a91
a29
a38a41a207
a67 a68a69 a70
a6
a74a8a7a8a78
a202
a56
a83a74a8a9
a109 a64a73a64a66a101a53a98a180a57a60a69a17a90a5a103a17a59a19a78a73a57a92a89a54a72a54a195
a71a79a57a60a89a41a59a48a64a117a68a74a96
a18
a21a114a207a24a23a76a72a74a78a73a59a149a112a43a68a41a101a17a69a40a103a87a59a23a103a12a68a41a69a103a44a45a46a33a48a47a6a49
a10
a10
a10
a10
a10
a27 a29
a30a19a18
a21a114a207a24a23a84a38a41a207a72a63
a56
a38
a202a58
a29 a20
a202
a18
a21a114a207 a202 a23
a10
a10
a10
a10
a10a12a11
a13
a21a26a18a11a44a57a47a23 a48
a88 a68a41a69a25a89a70a59a19a78a73a90a41a59a19a69a40a61a9a59a91a57a65a64 a3a40a22a75a39a27a18a125a96a32a72a41a64a66a71a16a15a4a15
a172a174a173a34a175a177a176a179a178a15a14
a109 a64a73a64a79a101a17a98a99a59a122a71a79a80a40a72a54a71a135a68a41a101a17a78a135a64a79a61a81a80a53a59a19a98a99a59a185a57a65a64a91a59a19a209a17a72a41a61a177a71a135a101a17a77a87a71a73a68a111a72a5a77a40a68a70a95a92a97a25a69a17a68a70a98a180a57a65a72a74a95a30a68a41a78a81a103a87a59a23a78 a25a18a113a188a60a118a135a191a117a80a40a72a54a71a149a98a99a59a48a72a74a69a53a64a149a62a153a59
a61a19a72a41a69a85a57a60a69a70a71a73a59a19a90a70a78a73a72a74a71a79a59a76a71a73a80a17a59a149a208a53a78a73a64a66a71a2a21a26a18 a44a16a47a23a46a71a73a59a19a78a73a98a99a64a128a57a60a69a85a71a73a80a17a57a60a64a117a191a30a72a27a97a25a95a92a68a70a78a128a64a79a59a19a78a73a57a92a59a48a64a58a59a19a209a25a77a40a72a74a69a53a64a79a57a92a68a70a69a85a59a9a209a17a72a70a61a177a71a79a95a60a97a41a118a143a191a117a80a17a59
a59a19a78a73a78a73a68a41a78a128a57a92a69a110a69a25a101a17a98a99a59a19a78a73a57a60a61a23a72a74a95a34a57a60a69a75a71a79a59a19a90a70a78a73a72a74a71a79a57a60a68a41a69
a71 a25 a27 a29
a30 a18
a21a114a207 a23a39a38a70a207 a63
a56
a38
a202a58
a29 a20
a202
a18
a21a114a207 a202a23a26a25
a47
a21a26a18a11a44a57a47a23 a48
a27 a29
a30
a33
a79
a91
a29
a18
a21a36a39a207a34a21a114a207 a23a84a23
a33
a207 a79
a91
a29
a207 a79
a91
a29
a38a70a207a89a94
a14a47a161a14a47a161a17a16 a11a194a22a25a22a75a20a23a29a51a28a7a49a174a20a48a26a7a22a102a13 a13a162a14a17a24a74a20a48a28a7a22a25a16a48a16 a164 a39a27a29a32a20a23a22a75a39a54a29a32a14
a165a43a166a75a167a169a168a47a170a21a171 a15
a93a143a209a17a72a41a61a9a71a79a69a17a59a48a64a79a64a63a61a19a68a41a69a53a103a17a57a206a71a73a57a92a68a70a69a111a78a73a59a23a116a75a101a17a57a60a78a79a59a48a64
a27 a29
a30
a14
a79
a21a114a207 a23a39a38a70a207 a25 a27 a29
a30
a21a38a7
a30
a44a13a7
a29
a207a57a44a13a7
a42
a207
a42
a44a11a10a12a10a36a10a55a44a9a7
a79
a207 a79a23a39a38a70a207 a25
a56
a38
a202a58
a29 a20
a202 a14
a79
a21a32a207 a202 a23
a96a114a68a41a78a76a72a41a69a25a97a5a64a66a59a19a71a63a68a74a96 a18a11a44 a47a67a61a9a68a25a59a19a18a107a61a19a57a92a59a23a69a75a71a73a64a17a7
a30
a33 a7
a29
a33a88a94a48a94a88a94a88a33 a7
a79
a13a15a31a34a33a30a29 a3a17a14a17a55a51a22a75a28a52a20a48a55a114a18
a27 a29
a30
a7
a30
a38a41a207a57a44 a27 a29
a30
a7
a29
a207a96a38a70a207 a44 a27 a29
a30
a7
a42
a207
a42
a38a70a207a57a44a35a10a36a10a12a10a58a44 a27 a29
a30
a7
a79
a207 a79a38a70a207 a25
a56
a38
a202a58
a29 a20
a202 a14
a79
a21a32a207 a202 a23
a172a174a173a34a175a177a176a179a178a43a178
a191a117a80a17a57a65a64a63a64a66a95a60a57a65a103a87a59a67a69a17a59a19a59a48a103a17a64a117a95a60a57a206a71a79a71a79a95a60a59a122a61a9a95a65a72a74a78a73a57a206a208a40a61a23a72a54a71a73a57a92a68a70a69a162a118 a19 a101a17a78a63a59a9a209a17a72a41a61a9a71a79a69a17a59a48a64a79a64a117a61a19a78a79a57a92a71a79a59a23a78a79a57a60a68a41a69a12a57a60a64
a27 a29
a30
a14
a79
a21a114a207 a23a39a38a70a207 a25 a27 a29
a30
a21a38a7
a30
a44a13a7
a29
a207a57a44a13a7
a42
a207
a42
a44a11a10a12a10a36a10a55a44a9a7
a79
a207 a79a23a39a38a70a207 a25
a56
a38
a202a58
a29 a20
a202 a14
a79
a21a32a207 a202 a23
a20
a62a63a80a17a57a65a61a81a80a111a57a65a64a117a71a73a80a17a59a122a64a79a72a41a98a99a59a135a72a70a64
a7
a30
a27 a29
a30
a38a41a207 a44 a7
a29
a27 a29
a30
a207a96a38a70a207a53a44 a10a12a10a36a10 a44 a7
a79
a27 a29
a30
a207 a79a38a70a207 a25a8a7
a30
a56
a38
a202a58
a29 a20
a202 a44 a7
a29
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a38
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a29 a20
a202 a207 a202 a44 a10a12a10a36a10a32a44 a7
a79
a56
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a29 a20
a202 a207 a202a79
a24a17a68a70a78a117a71a79a80a17a57a65a64a128a71a79a68a107a112a43a59a122a72a74a69a12a57a65a103a87a59a19a69a75a71a79a57a92a71a113a97a107a96a114a68a70a78a117a71a79a80a17a59 a21a38a18a11a44 a47a23a153a72a41a78a79a112a53a57a206a71a73a78a73a72a41a78a79a97a85a61a19a68a75a59a19a18a85a61a9a57a60a59a19a69a75a71a73a64a19a7 a202 a192a53a62a153a59a135a98a185a101a40a64a113a71a63a80a53a72a27a89a70a59
a71a79a80a53a59 a21a26a18a11a44a57a47a23a128a61a9a68a70a69a53a103a87a57a92a71a79a57a60a68a41a69a53a64
a56
a38
a202a58
a29 a20
a202a51a207a40a202
a2
a25a28a27 a29
a30
a207
a2
a38a70a207
a18a53a52a55a54
a9 a25a32a45a46a33a48a47a98a33a48a94a88a94a88a94a33 a18
a14a47a161a14a47a161a1a0 a11a194a22a25a22a75a20a23a29a51a28a7a49a174a20a48a26a7a22a102a13 a13a162a14a17a24a74a20a48a28a7a22a25a16a48a16 a164 a39a27a29a32a20a23a22a75a39a54a29a32a14
a165a43a166a75a167a169a168a47a170a21a171a98a19
a93a143a209a17a72a41a61a9a71a79a69a17a59a48a64a79a64a63a61a19a68a41a69a53a103a17a57a206a71a73a57a92a68a70a69a111a62a63a57a60a95a92a95a47a112a40a59a122a64a73a72a54a71a73a57a60a64a66a208a53a59a23a103a111a57a92a96a46a72a41a69a53a103a5a68a70a69a17a95a60a97a5a57a92a96
a27a16a29
a30
a38a70a207 a25
a56
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a29 a20
a202 a10 a47
a27 a29
a30
a207a24a38a41a207 a25
a56
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a29 a20
a202 a10a9a207 a202
a118
a118
a118
a27 a29
a30
a207 a79a38a41a207 a25
a56
a38
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a29 a20
a202 a10a9a207 a202a79
a14a47a161a14a47a161a3a2 a11a194a22a25a22a75a20a23a29a51a28a7a49a174a20a48a26a7a22a102a13 a13a162a14a17a24a74a20a48a28a7a22a25a16a48a16 a164 a39a27a29a32a20a23a22a75a39a54a29a32a14
a165a43a166a75a167a169a168a47a170a21a171a69a16
a6a63a59a23a68a41a78a73a90a70a72a74a69a53a57a92a183a23a57a92a69a17a90a181a59a9a209a17a72a41a61a9a71a79a69a17a59a48a64a79a64a117a59a48a116a75a101a53a72a54a71a73a57a92a68a70a69a53a64
SMA-HPC ?2000 MIT
Normalized 1-D
Problem
General Quadrature Scheme
Meeting the exactness
criteria
1
12 2
1
12
0
111 1
0n
lll l
nn
w
xx x w
xdxxx xw
#$#$#$
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&&’&&
%
a15 a37a43a28a7a55a51a29a32a28a30a22a75a14a17a39a75a192a87a64a79a57a92a69a40a61a9a59a67a207a40a202a79a188 a64a63a72a74a69a53a103
a20
a202a66a188 a64a63a72a74a78a73a59a135a101a17a69a17a197a25a69a17a68a54a62a63a69a53a64
a172a174a173a34a175a177a176a179a178 a17
a99a76a68a54a62 a62a63a80a40a72a54a71a135a57a60a64a135a72a85a77a53a78a73a72a70a61a177a71a79a57a65a61a19a72a41a95a7a62a117a72a27a97a110a68a74a96a153a61a19a68a41a98a99a77a17a101a87a71a73a57a92a69a53a90a5a71a73a80a17a59a181a59a23a89a54a72a74a95a60a101a53a72a54a71a73a57a92a68a70a69a36a77a40a68a70a57a92a69a75a71a81a64a15a72a41a69a53a103a36a62a128a59a19a57a60a90a41a80a75a71a81a64a28a27
a186a174a59a76a61a19a72a74a69a180a62a63a78a73a57a206a71a73a59a117a71a79a80a17a59a63a59a19a209a87a72a70a61a177a71a73a69a17a59a23a64a73a64a46a61a9a78a73a57a206a71a73a59a19a78a73a57a60a72a15a57a60a69a70a71a73a68a67a72a149a98a107a72a74a71a79a78a73a57a206a209a122a96a114a78a73a68a41a98a110a118a46a191a117a80a17a59a76a64a79a97a25a64a66a71a79a59a23a98a194a68a74a96a52a59a23a116a75a101a53a72a74a71a79a57a60a68a41a69a53a64
a57a65a64a117a69a17a68a74a71a76a59a48a72a41a64a79a97a99a71a79a68a85a64a79a68a41a95a60a89a41a59a135a64a66a57a60a69a53a61a9a59a67a207 a202 a188 a64a63a72a74a69a40a103
a20
a202 a188 a64a63a72a41a78a79a59a135a101a17a69a53a197a75a69a53a68a54a62a63a69a53a64a19a118
a210 a0
a14a47a161a14a47a161a1a0 a164 a37a43a42a45a44a30a33a7a20a23a29a51a28a7a49a50a20a23a26a30a22a6a33a149a37a40a29a51a28a52a20a23a16a99a14a17a28a30a86a35a0 a22a25a29a32a49a43a26a52a20a23a16
a165a43a166a75a167a169a168a47a170a21a171 a7
a88 a68a41a101a53a95a60a103a111a101a53a64a79a59 a15 a22 a3a99a20a48a37a40a28a3a2a92a16 a11a194a22a75a20a23a26a30a37a162a86
a4a90a21a6a5a46a23a26a25a57a45a8a7a10a9a12a11 a22 a5a14a13
a24
a22 a5a14a13
a91
a29
a63a15a5a14a13
a24
a25a8a63a16a4 a22 a5a14a13
a24
a191a117a80a17a59a135a69a17a68a70a69a17a95a92a57a60a69a17a59a48a72a74a78a117a96a114a101a17a69a53a61a9a71a79a57a60a68a41a69a12a96a114a68a41a78a12a99a91a59a19a62a117a71a73a68a41a69a110a57a60a64a128a71a73a80a17a59a19a69
SMA-HPC ?2000 MIT
Normalized 1-D
Problem
General Quadrature Scheme
Computing the points and
weights
1
12 2
1
12
0
111 1
0n
lll l
nn
w
xx x wwF
x
xdxxx xw
#$#$#$
%&%&%&
%&)*
%&=?=+,
%&-.%&%&
%&’(’(
%&’("
% &
% &
&&’&&
%
a172a174a173a34a175a177a176a179a178 a85
a99a76a59a23a62a117a71a79a68a70a69a162a188 a64a30a98a99a59a9a71a73a80a17a68a87a103a122a57a65a64a46a72a74a69a122a57a92a71a79a59a23a78a73a72a74a71a79a57a60a89a41a59a58a71a73a59a23a61a81a80a17a69a53a57a60a116a75a101a17a59a153a96a114a68a41a78a7a208a53a69a53a103a87a57a60a69a17a90a67a72a76a89a54a72a41a95a92a101a17a59a17a5a67a64a66a101a40a61a81a80a122a71a79a80a40a72a54a71a18a4a90a21a19a5a46a23a26a25
a45a17a118a46a191a117a80a17a59a76a98a99a59a19a71a79a80a17a68a87a103a107a57a65a64a58a112a53a72a70a64a66a59a48a103a180a68a41a69a107a95a60a57a92a69a17a59a48a72a74a78a73a57a92a183a23a57a92a69a53a90a135a71a79a80a17a59a76a77a53a78a79a68a70a112a17a95a92a59a23a98 a72a41a112a40a68a70a101a87a71a94a72a67a90a70a101a17a59a23a64a73a64a58a72a54a71a3a5a43a192a25a72a74a69a40a103a180a71a79a80a17a59a23a69
a101a17a77a52a103a17a72a54a71a73a57a92a69a53a90a102a71a79a80a17a59a111a89a54a72a74a95a60a101a17a59a85a68a74a96a20a5a125a112a25a97a174a64a79a68a41a95a60a89a25a57a92a69a53a90a110a71a79a80a17a59a85a95a60a57a60a69a17a59a23a72a41a78a79a57a60a183a19a59a48a103a125a77a17a78a79a68a70a112a17a95a60a59a19a98a110a118a102a104a105a69a45a77a53a72a74a78a79a71a79a57a65a61a9a101a53a95a60a72a41a78a23a192a7a71a79a80a17a59
a57a92a71a79a59a19a78a81a72a54a71a73a59a21a5 a13
a91
a29
a57a60a64a91a103a87a59a9a71a73a59a19a78a73a98a180a57a60a69a17a59a48a103a85a96a114a78a73a68a41a98a22a5 a13 a112a25a97a5a64a79a68a41a95a60a89a75a57a60a69a17a90a181a71a79a80a17a59a67a95a60a57a60a69a17a59a23a72a41a78a76a64a66a97a87a64a66a71a79a59a19a98a132a68a74a96a46a59a23a116a75a101a53a72a54a71a73a57a92a68a70a69a53a64
a4a90a21a6a5 a13 a23 a44a23a9 a11 a21a6a5 a13 a23a7a22a24a5 a13
a91
a29
a63a15a5 a13
a24
a25a57a45
a62a63a80a17a59a23a78a79a59a25a9a26a11 a21a19a5 a13 a23a107a57a65a64a107a71a73a80a17a59a28a27a75a72a41a61a19a68a41a112a17a57a65a72a74a69a10a101a114a98a181a101a17a95a206a71a73a57a60a103a17a57a92a98a99a59a19a69a40a64a66a57a60a68a41a69a53a72a41a95a15a103a17a59a19a78a73a57a92a89a54a72a54a71a73a57a92a89a70a59 a102a180a68a41a96a67a71a79a80a17a59a36a69a53a68a41a69a17a95a60a57a92a69a53a59a23a72a74a78
a96a114a101a17a69a53a61a9a71a79a57a60a68a41a69a29a4a90a21a19a5a46a23a9a118a134a191a117a80a17a59a102a57a206a71a73a59a19a78a81a72a54a71a73a57a92a68a70a69a21a57a65a64a107a61a9a68a70a69a70a71a73a57a92a69a25a101a17a59a48a103 a101a53a69a70a71a73a57a92a95a76a71a79a80a17a59a102a101a17a77a52a103a17a72a74a71a79a59a23a103a30a5a21a57a60a64a107a64a79a101 a18a85a61a9a57a60a59a19a69a75a71a79a95a60a97
a61a9a95a60a68a70a64a79a59a135a71a79a68a85a71a79a80a17a59a122a59a19a209a17a72a41a61a177a71a15a64a66a68a70a95a92a101a87a71a73a57a92a68a70a69a162a192a53a72a85a61a19a78a79a57a92a71a79a59a23a78a79a57a60a68a41a69a12a71a73a80a53a72a54a71a149a61a23a72a74a69a102a112a43a59a185a103a87a57a3a18a85a61a9a101a17a95a92a71a91a71a73a68a85a89a41a59a23a78a79a57a92a96a114a97a41a118a75a99a76a59a23a62a117a71a79a68a41a69a47a188 a64
a98a99a59a9a71a73a80a17a68a87a103a181a103a87a68a25a59a48a64a7a69a53a68a74a71a58a72a74a95a60a62a128a72a27a97a87a64a7a61a9a68a41a69a25a89a70a59a19a78a73a90a41a59a41a192a54a72a149a77a17a80a17a59a23a69a17a68a41a98a99a59a19a69a53a68a41a69a122a71a73a80a53a72a54a71a143a57a65a64a46a98a99a68a41a78a73a59a153a95a60a57a92a197a70a59a19a95a60a97a67a62a63a80a17a59a19a69a31a9 a11 a21a6a5 a23
a57a65a64a47a69a17a59a48a72a74a78a73a95a92a97a135a64a79a57a92a69a17a90a70a101a17a95a65a72a74a78a48a118 a24a17a68a70a78a47a98a99a68a41a78a73a59a153a72a41a112a40a68a70a101a87a71 a99a91a59a19a62a117a71a79a68a70a69a162a188 a64a47a98a99a59a9a71a73a80a17a68a87a103a34a192a74a64a79a59a19a59a58a71a73a80a17a59 a35 a118 a189a70a189 a35a33a32a25a210 a35 a118 a20a31a32 a0 a32 a32 a118 a0 a20a31a35
a61a9a68a70a101a17a78a81a64a66a59a15a69a17a68a74a71a73a59a23a64a23a118
a14a47a161a14a47a161a1a34 a164 a37a43a42a45a44a30a33a7a20a23a29a51a28a7a49a50a20a23a26a30a22a6a33a149a37a40a29a51a28a52a20a23a16a99a14a17a28a30a86a35a0 a22a25a29a32a49a43a26a52a20a23a16
a165a43a166a75a167a169a168a47a170a21a171 a37
a99a76a59a23a62a117a71a79a68a70a69a12a100a102a59a19a71a79a80a17a68a87a103a35a27a75a72a41a61a9a68a70a112a17a57a65a72a74a69a5a78a73a59a19a89a70a59a23a72a41a95a60a64a128a77a17a78a73a68a41a112a17a95a60a59a19a98
Normalized 1-D
Problem
General Quadrature Scheme
Computing the points and
weights
12 1 2
11
12 11 1
11 1 1 1 1
nn
F
ll l l l
xx x w w wwJ
x
xx xlwx lwx??
#$
%&)*
=+,
-.
’(
%%
&&’& & &’ &
%%%
Columns become
linearly dependent for
high order
2n
00 0
a172a174a173a34a175a177a176a179a178a49a26
a187a162a68a25a68a70a197a75a57a60a69a17a90a102a72a54a71a67a71a73a80a17a59a36a27a75a72a41a61a19a68a41a112a17a57a65a72a74a69a50a68a74a96a117a71a79a80a17a59a107a77a53a78a79a68a70a112a17a95a92a59a23a98a110a192a7a62a128a59a107a78a79a59a48a72a74a95a60a57a92a183a23a59a180a71a79a80a53a72a74a71a67a71a79a80a17a59a107a208a40a78a73a64a66a71 a25 a198a143a188a7a61a9a68a41a95a60a101a17a98a99a69a53a64
a112a43a59a23a61a9a68a70a98a99a59a143a57a92a69a53a61a19a78a79a59a48a72a41a64a79a57a92a69a53a90a41a95a60a97a76a95a92a57a60a69a17a59a48a72a74a78a73a95a92a97a15a103a87a59a19a77a43a59a19a69a53a103a17a59a19a69a75a71a162a96a114a68a41a78a47a95a60a72a41a78a79a90a70a59 a18a66a118a47a191a117a80a17a57a65a64a162a57a65a64a34a112a43a68a41a101a17a69a53a103a15a71a73a68a76a80a53a72a41a77a17a77a40a59a23a69a67a64a66a57a60a69a53a61a9a59
a210a41a210
a62a128a59a185a72a74a78a73a59a67a95a92a68a25a68a70a197a75a57a60a69a17a90a107a57a60a69a75a71a79a68a107a71a73a80a17a59a185a64a79a77a53a72a41a61a19a59
a56
a14 a7a75a198a1a0 a47a33a66a207a89a33a48a94a88a94a48a94a33a66a207 a79a3a2a70a118a63a191a117a80a17a57a65a64a76a112a53a72a70a64a66a57a65a64a91a72a41a95a92a62a117a72a27a97a87a64a117a112a40a59a48a61a9a68a41a98a99a59a48a64a76a57a60a95a92a95
a61a9a68a70a69a53a103a87a57a92a71a79a57a60a68a41a69a17a59a48a103a107a62a63a57a92a71a79a80a85a57a60a69a53a61a9a78a73a59a23a72a70a64a66a57a60a69a17a90a25a18a66a118a30a191a117a80a53a59a149a64a79a68a41a95a60a101a87a71a79a57a60a68a41a69a85a57a65a64a94a71a79a68a185a68a70a112a87a71a73a72a41a57a92a69a85a72a185a77a40a68a70a95a92a97a25a69a17a68a70a98a99a57a60a72a41a95a40a112a53a72a70a64a66a57a65a64a143a71a73a80a53a72a54a71
a57a65a64a30a22a177a69a17a68a70a78a79a98a107a72a74a95a60a57a60a183a19a59a23a103 a22a76a57a60a69a12a64a79a68a41a98a99a59a67a64a79a59a19a69a53a64a79a59a122a64a66a68a181a71a79a80a40a72a54a71a76a57a92a71a76a57a60a64a63a77a53a78a79a68a70a77a40a59a23a78a79a95a60a97a85a61a9a68a70a69a53a103a87a57a92a71a79a57a60a68a41a69a17a59a48a103a34a118
a14a47a161a14a47a161a32a163a5a4 a6 a16a48a22a43a15a12a29a8a7a94a22a75a39a54a22a25a28a52a20 a33a149a37a40a55a32a18a162a28a7a37a43a42a146a29a51a14a17a55a32a16
a165a43a166a75a167a169a168a47a170a21a171 a16
a93a143a209a17a72a41a61a9a71a79a69a17a59a48a64a79a64a63a61a19a78a79a57a92a71a79a59a23a78a79a57a65a72a181a62a63a57a92a95a60a95a162a112a43a59a122a64a79a72a74a71a79a57a65a64a113a208a53a59a48a103a5a57a92a96a143a72a74a69a53a103a12a68a41a69a53a95a92a97a85a57a92a96
Normalized 1-D
Problem
General Quadrature Scheme
Use different polynomials
() ( )
1
00
10
n
ii
i
cxdx wcx
=
=!"
() ( )
1
11
10
n
ii
i
cxdx wcx
=
=!"
() ( )
1
10
n
lili
i
cxdx wcx
=
=!"
Each polynomial mustic
Contain an termix
Be linearly independent
BUT
a172a174a173a34a175a177a176a179a178 a11
a191a117a80a17a59a99a68a41a69a53a95a92a97a36a103a87a57a92a182a52a59a19a78a73a59a19a69a53a61a19a59a185a96a114a78a73a68a41a98 a71a73a80a17a59a99a77a17a78a73a59a19a89a25a57a92a68a70a101a53a64a135a64a66a59a19a71a15a68a41a96a128a61a19a78a79a57a92a71a79a59a23a78a79a57a65a72a5a57a65a64a149a71a79a80a53a72a74a71a135a71a79a80a17a59a48a64a66a59a99a77a43a68a41a95a60a97a75a69a53a68a41a98a99a57a60a72a41a95a60a64
a80a53a72a27a89a70a59a149a112a43a59a9a71a79a71a79a59a19a78a91a77a17a78a79a68a70a77a40a59a23a78a66a71a73a57a92a59a48a64a128a71a79a80a53a72a41a69a5a71a73a80a17a59a67a68a41a69a53a59a23a64a117a62a128a59a135a61a81a80a53a68a70a64a79a59a15a112a43a59a9a96a114a68a70a78a79a59a70a118
a14a47a161a14a47a161a32a163a43a163 a9 a39a27a20a23a26a30a37a43a49a40a37a43a28a7a14a53a55a17a33a149a37a40a55a32a18a162a28a7a37a43a42a146a29a32a14a53a55a32a16
a165a43a166a75a167a169a168a47a170a21a171 a18
a24a17a68a70a78a117a71a79a80a17a59a67a69a53a68a41a78a73a98a99a72a41a95a92a57a60a183a19a59a48a103a85a57a60a69a75a71a79a59a19a90a70a78a73a72a41a95a51a192a75a71a113a62a128a68a99a77a40a68a54a97a25a69a17a68a70a98a180a57a65a72a74a95a65a64a117a72a74a78a73a59a67a64a79a72a41a57a60a103a5a71a79a68a99a112a43a59a122a37a53a39a27a20a23a26a30a37a43a49a40a37a43a28a7a14a53a55a7a57a92a96
a27 a29
a30a11a10
a202 a21a114a207 a23
a10
a2 a21a32a207a24a23a84a38a41a207 a25a32a45
a18a53a52a55a54
a9
a27
a25a134a203
a191a117a80a17a59a110a72a74a112a43a68a54a89a41a59a111a57a92a69a75a71a79a59a23a90a41a78a81a72a74a95a153a57a60a64a180a68a74a96a169a71a73a59a19a69 a78a79a59a19a96a114a59a19a78a73a78a79a59a48a103a174a71a79a68a125a72a41a64a99a72a74a69a21a57a92a69a53a69a17a59a19a78a99a77a17a78a73a68a87a103a87a101a53a61a177a71a107a72a74a69a40a103a146a72a41a64a73a61a9a78a73a57a92a112a43a59a23a103a146a71a79a80a17a59
a69a17a68a41a71a73a72a54a71a73a57a92a68a70a69
a21
a10
a202a53a33
a10
a2 a23a26a25a28a27a16a29
a30 a10
a202a84a21a32a207a24a23
a10
a2a65a21a114a207 a23a39a38a70a207
a191a117a80a17a59a99a61a9a68a70a69a17a69a17a59a23a61a9a71a79a57a60a68a41a69a50a112a40a59a19a71a113a62a153a59a23a59a19a69a179a77a43a68a41a95a60a97a25a69a17a68a41a98a99a57a65a72a74a95a30a57a92a69a53a69a17a59a19a78a135a77a17a78a73a68a87a103a87a101a53a61a177a71a81a64a135a72a74a69a53a103a179a89a41a59a23a61a9a71a79a68a70a78a149a57a60a69a17a69a17a59a23a78a15a77a53a78a79a68a87a103a87a101a53a61a9a71a73a64
a61a19a72a41a69a111a112a43a59a122a64a79a59a19a59a19a69a12a112a25a97a5a64a73a72a74a98a99a77a17a95a60a57a92a69a17a90a40a118
a14a47a161a14a47a161a32a163a17a160 a13a10a13a34a44a46a55a32a37a43a29a114a20a48a29a32a28a30a49a179a20a48a26a7a22 a15a111a29a12a7a94a22a75a39a27a22a25a28a52a20a36a33a91a37a43a55a114a18a162a28a30a37a40a42 a29a32a14a17a55a51a16
a165a43a166a75a167a169a168a47a170a21a171 a4
a88 a68a41a69a40a64a66a57a65a103a87a59a19a78a117a78a73a59a19a62a63a78a73a57a206a71a73a57a92a69a53a90a181a71a73a80a17a59a67a59a9a209a17a72a70a61a177a71a79a69a53a59a23a64a73a64a128a61a19a78a79a57a92a71a79a59a23a78a79a57a65a72
Normalized 1-D
Problem
General Quadrature Scheme
Exploiting the different
polynomials
() ( )
1
00
10
n
ii
i
cxdx wcx
=
=!"
() ( )
1
11
10
Low order terms
n
nii
i
cxdx wcx??
=
!"
!"""""#"""""$
() ( )
1
10
n
nini
i
cxdx wcx
=
=!"
() ( )
1
21 21
10
High Order Terms
n
nini
i
cxdx wcx??
=
!
!"""""#"""""$
a210 a32
a6a63a59a23a61a23a72a74a95a60a95a52a71a79a80a53a72a74a71a30a18a84a21a1a0 a14
a52
a18a6a5
a56
a23a4a25 a23 a198a21a63a16a47a65a21a1a0
a52 a18 a10 a52a69a60a69a18 a18
a203
a10
a203
a60
a198a3a2
a56
a23
a172a174a173a34a175a177a176a179a178 a18
a186a174a59a17a16a113a101a40a64a113a71a76a61a19a72a41a95a92a95a43a71a73a80a17a59a15a208a53a78a73a64a66a71a2a21a114a198 a63 a47a23a153a61a9a68a41a69a40a103a87a57a206a71a73a57a92a68a70a69a53a64a117a72a41a64a153a71a79a80a53a59 a22a9a95a92a68a54a62a128a59a19a78a153a68a41a78a81a103a87a59a19a78a153a71a79a59a23a78a79a98a107a64 a22a91a72a41a69a53a103a5a71a79a80a17a59a15a95a60a72a70a64a113a71
a198a50a61a19a68a41a69a53a103a17a57a206a71a73a57a92a68a70a69a53a64a63a72a41a64a128a71a73a80a17a59 a22a177a80a17a57a60a90a41a80a53a59a19a78a117a68a41a78a81a103a87a59a19a78a117a71a73a59a19a78a73a98a107a64 a22a74a118
a14a47a161a14a47a161a32a163 a14 a13a10a13a34a44a46a55a32a37a43a29a114a20a48a29a32a28a30a49a179a20a48a26a7a22 a15a111a29a12a7a94a22a75a39a27a22a25a28a52a20a36a33a91a37a43a55a114a18a162a28a30a37a40a42 a29a32a14a17a55a51a16
a165a43a166a75a167a169a168a47a170a16a15 a31
a88 a72a74a69a12a62a63a78a73a57a206a71a73a59a149a71a73a80a17a59a67a80a17a57a60a90a41a80a17a59a23a78a63a68a41a78a81a103a87a59a19a78a128a71a73a59a19a78a73a98a99a64a91a103a87a57a206a182a52a59a19a78a73a59a19a69a75a71a73a95a92a97
a5
a29a30
a10
a56
a21a32a207a24a23a84a38a41a207a86a25 a54 a56
a202a58
a29
a20
a202
a10
a56
a21a32a207a40a202a52a23 a7
a5
a29a30
a10
a56
a21a114a207a24a23
a10
a30
a21a114a207 a23a39a38a70a207 a25 a54 a56
a202a58
a29
a20
a202
a10
a56
a21a114a207a40a202a77a23
a10
a30
a21a32a207a40a202a77a23
a118
a118
a118
a118
a118
a118
a5
a29a30
a10
a42
a56
a11
a29
a21a32a207a24a23a39a38a70a207 a25 a54 a56
a202a58
a29
a20
a202
a10
a42
a56
a11
a29
a21a114a207a43a202a23 a7
a5
a29a30
a10
a56
a21a114a207 a23
a10
a56
a11
a29
a21a114a207a24a23a84a38a41a207a86a25
a54a57a56
a202a58
a29
a20
a202
a10
a56
a21a114a207a43a202a23
a10
a56
a11
a29
a21a32a207a40a202a52a23
a191a117a80a17a59a135a77a17a78a73a68a87a103a87a101a53a61a177a71a81a64
a10
a56
a21a32a207a24a23
a10
a2a65a21a114a207 a23a117a72a74a78a73a59a15a95a60a57a60a69a17a59a23a72a41a78a79a95a60a97a85a57a60a69a53a103a87a59a19a77a43a59a19a69a40a103a87a59a19a69a75a71a16a15
a172a174a173a34a175a177a176a179a178 a30
a104a105a69a102a71a73a80a17a57a60a64a15a64a79a95a92a57a65a103a87a59a180a62a153a59a122a59a19a209a87a77a17a78a79a59a48a64a79a64a76a71a79a80a17a59 a22a177a80a53a57a92a90a70a59a19a78a76a68a70a78a73a103a87a59a23a78a76a71a73a59a19a78a73a98a99a64 a22a122a72a70a64a91a61a19a68a41a69a53a103a87a57a92a71a79a57a60a68a41a69a40a64a91a57a60a69a25a89a41a68a41a95a60a89a25a57a92a69a53a90 a22a177a95a60a68a54a62a153a59a23a78
a68a41a78a81a103a87a59a23a78a153a71a73a59a19a78a73a98a107a64 a22a74a118
a14a47a161a14a47a161a32a163 a2 a6 a16a48a29a32a28a30a49 a9 a39a27a20a23a26a30a37a43a49a40a37a43a28a7a14a53a55a32a29a114a20a81a18 a14a17a28a7a86a5a4a111a37a162a37a40a20a23a16
a165a43a166a75a167a169a168a47a170a16a15a34a171
a4a91a64a79a59a135a68a41a78a79a71a79a80a17a68a70a90a41a68a70a69a53a72a74a95a52a77a43a68a41a95a60a97a75a69a53a68a41a98a99a57a60a72a41a95a60a64
SMA-HPC ?2000 MIT
Normalized 1-D
Problem
General Quadrature Scheme
Using orthogonality and
roots
() () ( ) ( )
1
11
10
n
nn inini
i
cxc xdx wcxc x??
=
!"
() () ( ) ( )
1
00
10
n
nii
i
cxcxdx wcxcx
=
=!"
0
0 0
0
a56a58a57a65a61a81a197a107a71a79a80a53a59a67a207a40a202a79a188 a64a117a71a73a68a99a112a40a59a67a198a174a78a79a68a25a68a41a71a73a64a128a68a74a96
a10
a56
a21a114a207 a23
a191a117a80a17a59a135a80a17a57a60a90a41a80a17a59a23a78a63a68a41a78a81a103a87a59a19a78a63a61a19a68a41a69a53a64a66a71a79a78a81a72a74a57a60a69a75a71a73a64a117a72a74a78a73a59a135a59a9a209a17a72a41a61a9a71a79a95a60a97a5a64a79a72a74a71a79a57a65a64a113a208a40a59a23a103 a15
a172a174a173a34a175a177a176a179a178 a22
a191a117a80a17a57a65a64a99a59a19a95a60a59a19a90a75a72a74a69a75a71a99a64a66a71a79a59a23a77a21a78a73a59a19a95a60a57a60a59a23a64a99a68a41a69 a77a43a68a41a95a60a97a75a69a53a68a41a98a99a57a60a72a41a95a117a68a41a78a79a71a79a80a17a68a70a90a41a68a41a69a40a72a74a95a60a57a206a71a113a97a70a118 a104a38a96a149a62a128a59a102a61a81a80a17a68a25a68a70a64a79a59a5a71a73a80a17a59a102a77a43a68a41a95a60a97a70a195
a69a17a68a70a98a180a57a65a72a74a95
a10
a56
a21a114a207 a23a107a64a66a101a40a61a81a80a21a71a73a80a53a72a54a71a111a57a206a71a5a57a65a64a107a68a70a78a66a71a73a80a17a68a41a90a70a68a41a69a53a72a41a95a153a71a73a68 a72a74a95a60a95a76a77a43a68a41a95a60a97a25a69a17a68a41a98a99a57a65a72a74a95a65a64a99a68a74a96a67a57a92a69a17a96a114a59a19a78a73a57a92a68a70a78a85a103a87a59a19a90a70a78a79a59a23a59
a101a114a57a200a118 a59a41a118
a10
a30
a21a32a207a24a23a6a33
a10
a29
a21a114a207 a23a33a88a94a48a94a88a94a33
a10
a56
a11
a29
a21a32a207a24a23a53a102a122a72a74a69a40a103a174a71a79a80a17a59a85a207 a202 a188 a64a185a72a41a78a79a59a107a78a73a68a25a68a74a71a81a64a67a68a74a96a117a71a79a80a53a57a60a64a185a77a40a68a70a95a92a97a25a69a17a68a70a98a180a57a65a72a74a95a200a192a7a71a79a80a17a59a23a69
a210 a189
a71a79a80a53a59a12a80a53a57a92a90a70a80a17a59a19a78a99a68a41a78a81a103a87a59a23a78a180a198 a61a9a68a41a69a40a103a87a57a206a71a73a57a92a68a70a69a53a64a107a72a74a78a73a59a110a72a74a101a87a71a73a68a41a98a107a72a54a71a73a57a60a61a23a72a74a95a60a95a92a97a45a64a73a72a54a71a79a57a65a64a66a208a53a59a23a103a34a118 a99a76a68a41a71a79a59a12a71a79a80a53a72a74a71a99a96a114a68a41a78a99a71a79a80a53a57a60a64
a103a87a59a23a78a79a57a60a89a54a72a54a71a79a57a60a68a41a69a180a62a128a59a63a101a53a64a66a59a48a103a180a77a40a68a70a95a92a97a25a69a17a68a70a98a180a57a65a72a74a95a65a64a46a62a63a80a17a57a65a61a81a80a107a72a74a78a73a59a153a68a70a78a66a71a73a80a17a68a41a90a70a68a41a69a53a72a41a95a75a68a70a69a180a71a79a80a17a59a63a57a60a69a75a71a79a59a19a78a73a89a54a72a74a95a89a44a45a46a33a48a47a6a49a38a118 a115 a101a40a61a81a80
a77a43a68a41a95a60a97a75a69a53a68a41a98a99a57a60a72a41a95a60a64a117a72a41a78a79a59a135a64a66a80a53a57a206a96a169a71a73a59a23a103a110a72a74a69a53a103a12a64a73a61a19a72a74a95a60a59a23a103a5a89a41a59a23a78a73a64a79a57a92a68a70a69a53a64a153a68a74a96a30a71a79a80a17a59a67a61a9a95a65a72a41a64a73a64a79a57a60a61a23a72a74a95a34a187a162a59a23a90a41a59a19a69a40a103a87a78a79a59a15a77a43a68a41a95a60a97a75a69a53a68a41a98a99a57a206a195
a72a74a95a65a64a23a192a87a62a63a80a17a57a60a61a81a80a110a72a74a78a73a59a135a68a41a78a79a71a79a80a17a68a70a90a41a68a70a69a53a72a74a95a43a68a41a69a111a71a79a80a17a59a67a57a60a69a75a71a79a59a23a78a79a89a54a72a74a95a26a44a63a2a47a98a33a48a47a6a49a38a118
a14a47a161a14a47a161a32a163 a16 a2a162a14a17a20a23a29a32a16a48a35a200a18a162a29a32a28a30a49a174a20a23a26a30a22a1a0a94a37a4a3a185a22a75a39 a9 a39a54a86a7a22a75a39 a164 a37a43a28a7a16a23a20a19a39a54a14a17a29a51a28a52a20a23a16
a165a43a166a75a167a169a168a47a170a16a15 a15
a109 a69a102a72a41a112a17a112a17a78a73a59a19a89a25a57a60a72a74a71a79a59a48a103a85a59a9a209a17a72a70a61a177a71a79a69a53a59a23a64a73a64a117a59a23a116a75a101a53a72a54a71a73a57a92a68a70a69
a2a3
a4
a10
a30
a21a114a207
a29
a23 a10a12a10a36a10
a10
a30
a21a114a207
a56
a23
a118
a118
a118
a118
a118
a118
a118
a118
a118
a10
a56
a11
a29
a21a32a207
a29
a23 a10a12a10a36a10
a10
a56
a11
a29
a21a114a207
a56
a23
a5a7a6
a8
a2a3
a4 a20
a29
a118
a118
a118
a20
a56
a5a7a6
a8
a25
a2a3
a4
a5
a29a30
a10
a30
a21a114a207 a23a39a38a70a207
a118
a118
a118
a5
a29a30
a10
a56
a11
a29
a21a32a207a24a23a39a38a70a207
a5a7a6
a8
a99a76a68a54a62 a55a32a29a51a28a7a22a25a14a87a39a25a192a87a207a43a202a66a188 a64a76a72a74a78a73a59a15a197a75a69a53a68a54a62a63a69
a6a63a68a54a62a76a64a117a72a74a78a73a59a67a64a79a72a41a98a99a77a17a95a92a59a48a103a5a68a70a78a66a71a73a80a17a68a41a90a70a68a41a69a53a72a41a95a40a77a43a68a41a95a60a97a25a69a17a68a41a98a99a57a65a72a74a95a65a64a19a118
a172a174a173a34a175a177a176a179a178 a41
a9a128a97a50a101a40a64a66a57a60a69a17a90a36a71a73a80a17a59a5a78a73a68a25a68a74a71a81a64a122a68a74a96
a10
a56
a21a114a207a24a23a122a96a114a68a41a78a181a71a79a80a53a59a5a207a40a202a79a188 a64a19a192a46a71a73a80a17a59a5a80a17a57a60a90a41a80a53a59a19a78a180a68a41a78a81a103a87a59a19a78a185a61a19a68a41a69a53a64a66a71a79a78a81a72a74a57a60a69a75a71a73a64a181a72a74a78a73a59a5a72a74a101a87a195
a71a79a68a70a98a107a72a54a71a79a57a65a61a19a72a41a95a92a95a60a97a36a64a73a72a54a71a79a57a65a64a66a208a53a59a23a103a34a118 a115 a57a60a69a53a61a19a59a181a71a73a80a17a59a107a207a40a202a79a188 a64a122a72a74a78a73a59a181a69a53a68a54a62 a197a25a69a17a68a54a62a63a69a162a192a34a68a70a69a17a95a92a97a36a71a79a80a17a59a107a62a128a59a19a57a60a90a41a80a75a71a73a64a135a72a41a78a79a59a107a64a66a71a79a57a60a95a92a95
a101a17a69a17a197a25a69a17a68a54a62a63a69a47a118a91a191a117a80a17a59a122a95a60a68a54a62a153a59a23a78a91a198a45a61a9a68a70a69a53a64a66a71a79a78a81a72a74a57a60a69a70a71a81a64a91a61a23a72a74a69a102a112a43a59a185a101a53a64a79a59a23a103a110a71a73a68a111a103a87a59a19a71a79a59a23a78a79a98a99a57a60a69a17a59a67a71a79a80a53a59a185a62a128a59a19a57a60a90a41a80a75a71a73a64a23a192a40a90a70a59a19a69a87a195
a59a19a78a81a72a54a71a73a57a92a69a53a90a180a72a180a95a60a57a92a69a17a59a48a72a74a78a76a64a79a97a87a64a113a71a73a59a19a98a110a118
a13a117a138a11a10 a17a84a141a46a147a149a142a25a142a75a144a66a141a143a148 a7a146a147a91a141a10a9a76a158a17a141a30a151a17a147a76a158 a5a15a0a63a147a13a1 a1 a141a46a158a13a12
a14a47a161a3a2a7a161a32a163 a5a111a55a32a49a43a37a53a39a54a29a114a20a48a26a7a42 a2a34a20a48a22a70a44a46a16
a165a43a166a75a167a169a168a47a170a16a15a46a19
a210 a118 a88 a68a70a69a53a64a66a71a79a78a73a101a53a61a177a71a63a198 a44 a47a135a68a70a78a66a71a73a80a17a68a41a90a70a68a41a69a53a72a41a95a40a77a43a68a41a95a60a97a25a69a17a68a41a98a99a57a65a72a74a95a65a64
a27 a29
a30 a10
a202a84a21a32a207a24a23
a10
a2a65a21a114a207 a23a39a38a70a207 a25a32a45
a18a53a52a55a54
a9
a27
a25a134a203
a32 a118 a88 a68a70a98a99a77a17a101a87a71a79a59a91a198a111a39a54a37a162a37a40a20a23a16a70a192a74a207a40a202a53a33a58a203 a25 a47a98a33a48a94a88a94a48a94a6a33a66a198a12a68a74a96a43a71a79a80a17a59a91a198a34a204a32a205a135a68a41a78a81a103a87a59a19a78a143a68a70a78a66a71a73a80a17a68a41a90a70a68a41a69a53a72a41a95a25a77a40a68a70a95a92a97a25a69a17a68a70a98a180a57a65a72a74a95
a64a79a101a53a61a81a80a12a71a79a80a40a72a54a71
a10
a56
a21a114a207a43a202a77a23a26a25a57a45
a189a53a118 a115 a68a41a95a60a89a41a59a67a72a180a95a92a57a60a69a17a59a23a72a41a78a76a64a66a97a87a64a66a71a79a59a23a98 a96a114a68a41a78a117a71a73a80a17a59 a3a185a22a25a29a32a49a43a26a52a20a23a16
a20
a202
a74 a118 a109 a77a17a77a17a78a73a68a27a209a87a57a92a98a107a72a54a71a73a59a15a71a79a80a17a59a67a57a60a69a75a71a79a59a19a90a70a78a73a72a41a95a34a72a41a64a63a72a99a64a66a101a53a98
a5
a29a30
a18
a21a32a207a24a23a39a38a70a207 a25a55a54 a56
a202a58
a29
a20
a202
a18
a21a114a207 a202a23
a172a174a173a34a175a177a176a18a17 a14
a191a117a80a17a57a65a64a67a64a66a95a60a57a65a103a87a59a107a64a66a101a17a98a99a98a107a72a74a78a73a57a60a183a19a59a23a64a15a71a73a80a17a59a180a71a79a59a48a61a81a80a17a69a17a57a65a116a70a101a53a59a107a68a74a96a94a208a40a69a53a103a87a57a60a69a17a90a102a62a153a59a23a57a92a90a70a80a75a71a73a64a135a72a74a69a53a103a50a57a60a69a70a71a73a59a19a90a70a78a73a72a74a71a79a57a60a68a41a69a179a77a43a68a41a57a60a69a75a71a73a64
a96a114a68a41a78a10a4a135a72a41a101a53a64a73a64a63a116a70a101a40a72a41a103a87a78a81a72a54a71a73a101a17a78a79a59a70a118
a210 a74
a14a47a161a3a2a7a161a65a160 a5a99a24a41a24a70a33a47a39a54a14a17a24a41a18 a4a5a22a25a16a48a33a7a55a32a20
a165a43a166a75a167a169a168a47a170a16a15a30a16
a27 a29
a30 a18
a21a32a207a24a23a84a38a41a207 a20
a56
a38
a202a58
a29 a20
a202
a18
a21a32a207a40a202a52a23
a0 a59a19a97a85a77a53a78a79a68a70a77a40a59a23a78a66a71a73a57a92a59a48a64a117a68a74a96a47a71a73a80a17a59a67a98a99a59a9a71a73a80a17a68a87a103
a0
a109 a69a99a198a52a195a200a77a43a68a41a57a60a69a75a71 a4a135a72a41a101a53a64a73a64a143a116a70a101a40a72a41a103a87a78a81a72a54a71a73a101a17a78a79a59a153a78a79a101a53a95a92a59a117a57a65a64a58a22a31a13a162a14a17a24a74a20a117a96a114a68a70a78a143a77a40a68a70a95a92a97a25a69a17a68a70a98a180a57a65a72a74a95a65a64a30a68a74a96a52a68a41a78a81a103a87a59a19a78 a23 a198a4a63 a47
a0
a93a94a78a79a78a73a68a41a78a117a57a65a64a63a77a17a78a73a68a41a77a43a68a41a78a79a71a79a57a60a68a41a69a53a72a41a95a40a71a73a68 a22
a29
a42
a56
a24
a42
a56
a13a117a138a2a1 a0a117a144a2a1 a156a4a3a6a5a8a7 a147a76a141a10a9a76a158a17a141a30a151a17a147a76a158a11a5a12a0a63a150a53a159a13a5a14a1a15a5
a14a47a161a17a16a162a161a32a163 a82 a22a25a28a30a22a70a39a54a14a17a55a117a28a32a31a34a33a149a37a40a29a51a28a52a20a36a35a7a37a53a39a54a42a174a33a7a55a51a14 a165a43a166a75a167a169a168a47a170a16a15 a7
a27 a29
a30 a18
a21a32a207a24a23a39a38a70a207a61a20
a56
a38
a202a58
a29
a47
a198
a18
a22 a203 a63
a29
a42
a198 a24
SMA-HPC ?1999 MIT
Normalized 1-D
Problem
Simple Quadrature Scheme
()f x
x0 1
a0 a59a19a97a5a77a17a78a79a68a70a77a40a59a23a78a66a71a113a97a107a68a41a96a7a71a73a80a17a59a67a98a99a59a9a71a73a80a17a68a87a103
a0
a93a94a78a79a78a73a68a41a78a117a57a65a64a63a77a17a78a73a68a41a77a43a68a41a78a79a71a79a57a60a68a41a69a53a72a41a95a40a71a73a68
a29
a56
a14
a18a20a19a22a21
a176a24a23a26a25a28a27a30a29a41a176 a17a50a190a91a68a12a97a41a68a70a101a50a64a79a59a19a59a122a71a73a80a53a72a54a71a135a71a73a80a17a59a180a64a79a57a60a98a180a77a53a95a92a59a99a116a75a101a53a72a41a103a87a78a81a72a54a71a73a101a17a78a73a59a181a64a73a61a81a80a17a59a19a98a99a59a180a57a60a64a135a72a5a64a79a77a43a59a23a61a9a57a65a72a74a95a143a61a23a72a41a64a79a59
a68a74a96a1a4a135a72a74a101a53a64a73a64a63a116a75a101a53a72a41a103a17a78a73a72a74a71a79a101a17a78a73a59 a27
a13a117a138a2a3 a4a50a152 a1 a156a149a141a30a158a40a144a66a148a6a5 a0a117a144a2a1 a156a4a3a6a5a19a7a146a147a91a141a10a9 a141a143a148a13a9 a17a84a141a46a147a149a142a25a142 a7 a147a76a141a10a9
a165a43a166a75a167a169a168a47a170a16a15 a37
SMA-HPC ?1999 MIT
Gauss-Quad
Simple Quad
Number of Points
E
r
r
o
r
Approximately
Integrating
1
1
cos2 xdxpi
?
"
a172a174a173a34a175a177a176a18a17a47a178
a99a76a68a41a71a79a57a65a61a9a59a15a71a79a80a40a72a54a71a63a96a114a68a41a78a76a72a99a64a79a98a99a68a75a68a41a71a79a80a12a96a114a101a17a69a53a61a9a71a79a57a60a68a41a69
a18
a21a114a207 a23 a25
a10 a52 a56
a21 a23a8a7 a207 a23a177a192a53a62a63a80a53a57a60a61a81a80a110a57a65a64a117a57a92a69a87a208a40a69a17a57a206a71a73a59a19a95a60a97a111a103a87a57a92a182a52a59a19a78a73a59a19a69a87a195
a71a79a57a65a72a74a112a53a95a92a59a70a192 a4a135a72a74a101a53a64a73a64a63a116a75a101a53a72a41a103a87a78a81a72a54a71a73a101a17a78a73a59a91a96a32a72a41a78a63a68a41a101a87a71a73a77a40a59a23a78a66a96a114a68a70a78a79a98a107a64a128a71a79a80a53a59a122a64a66a57a60a98a99a77a17a95a92a59a122a116a75a101a53a72a70a103a87a78a81a72a54a71a79a101a53a78a79a59a135a64a79a61a81a80a53a59a19a98a99a59
a210a28a39
a13a117a138a1a0 a4a50a152 a1 a156a149a141a30a158a40a144a66a148a6a5 a0a117a144a2a1 a156a4a3a6a5a19a7a146a147a91a141a10a9 a141a143a148a13a9 a17a84a141a46a147a149a142a25a142 a7 a147a76a141a10a9
a14a47a161a3a2a162a161a32a163 a13a22a3a17a14a17a55a32a33a30a14a87a20a48a29a32a37a43a28 a33a91a37a43a29a32a28a52a20 a33a67a55a32a14a17a24a70a22a25a42a146a22a25a28a52a20 a165a43a166a75a167a169a168a47a170a16a15 a16
SMA-HPC ?1999 MIT
Symmetrically Normalized 1-D
Problem
Comparing Quadratures
Evaluation Point Placement
-1 1
Simple-Quadrature Points
0
Gauss-Quadrature Points
0-1 1
a99a76a68a41a71a79a57a65a61a9a59a15a71a79a80a53a59a122a61a9a95a60a101a53a64a113a71a73a59a19a78a73a57a92a69a53a90a99a72a74a71a63a57a92a69a75a71a79a59a23a78a79a89a54a72a41a95a34a59a19a69a53a103a17a64
a172a174a173a34a175a177a176a18a17 a17
a104a105a69a146a71a73a80a17a59 a4a135a72a74a101a53a64a73a64a99a116a75a101a53a72a41a103a87a78a81a72a54a71a73a101a17a78a73a59a111a64a73a61a81a80a17a59a19a98a99a59a85a71a73a80a17a59a110a59a19a89a54a72a74a95a60a101a53a72a74a71a79a57a60a68a41a69 a77a43a68a41a57a60a69a75a71a73a64a99a72a74a78a73a59a12a78a79a68a25a68a74a71a81a64a185a68a41a96a149a187a162a59a23a90a41a59a23a69a53a103a87a78a73a59
a77a43a68a41a95a60a97a75a69a53a68a41a98a99a57a60a72a41a95a60a64a128a62a63a80a17a57a65a61a81a80a102a72a74a78a73a59a15a61a19a95a92a101a40a64a113a71a73a59a19a78a73a59a23a103a110a72a54a71a117a71a79a80a53a59a67a59a19a69a53a103a17a64a117a68a41a96a7a71a73a80a17a59a67a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95a200a118
a2 a3a5a4 a11a7a6a107a8a9a10a9a8a185a2a111a6a9a124 a1a11a10 a11a75a1a30a10a12a11a63a6 a9a10a1a46a136a2a11a12a6a19a11a12a3
a10a76a138a66a139 a13a13a12a15a14a122a141a46a156a4a3a66a141a143a150a53a5a17a16a19a18a63a141a10a1 a156a4a3a6a5
a2a7a161a32a163a43a161a32a163 a164 a14a17a55a51a24a41a33a30a55a32a14a87a20a48a29a32a28a30a49a125a20a23a26a30a22a21a20 a2a47a22a25a55a32a35 a31 a106 a22a75a39a54a42a22a20
a165a43a166a75a167a169a168a47a170a16a15 a18
3-D Laplace’s
Equation
Panel i
ic
x Collocationpoint
,
0
ii
ii
cc
Panel AreaA
xx?≈"#$#%One point quadrature
Approximation
x
y
z
, is an integrable singularity
1
i
ii
panel i cxx
AdS′?=!
Basis Function Approach
,
1
i
ii
pa cnel i xx
A dS′′?= !
a172a174a173a34a175a177a176a18a17 a85
a99a76a68a54a62a199a95a92a59a19a71a73a64a117a90a41a68a181a112a53a72a41a61a81a197a99a71a73a68a99a68a41a101a17a78a117a77a17a78a73a68a41a112a17a95a60a59a19a98 a68a41a96a7a64a79a68a41a95a60a89a75a57a60a69a17a90a99a187a47a72a41a77a17a95a60a72a70a61a9a59a70a188 a64a153a59a23a116a75a101a53a72a74a71a79a57a60a68a41a69a5a68a70a69a12a72a180a189a41a190 a103a17a68a41a98a107a72a74a57a60a69
a101a53a64a79a57a92a69a53a90a146a112a40a68a70a101a17a69a53a103a17a72a41a78a79a97 a57a60a69a70a71a73a59a19a90a70a78a73a72a41a95a91a78a73a59a19a77a17a78a73a59a23a64a79a59a19a69a75a71a73a72a74a71a79a57a60a68a41a69a162a118 a186a174a59a179a78a73a59a23a72a41a95a92a57a60a183a19a59a36a71a79a80a53a72a74a71a111a62a128a59a179a69a17a68a54a62 a80a53a72a27a89a70a59a179a64a66a68a70a98a99a59
a64a79a68a41a77a17a80a17a57a65a64a66a71a79a57a65a61a19a72a54a71a73a59a23a103a111a71a79a68a25a68a70a95a60a64a76a71a79a68a85a80a53a72a41a69a53a103a87a95a60a59a122a57a60a69a70a71a73a59a19a90a70a78a73a72a41a95a60a64a117a68a41a96a30a96a114a101a53a69a53a61a177a71a73a57a92a68a70a69a53a64a63a71a73a80a53a72a54a71a135a72a74a78a73a59a122a64a66a98a99a68a25a68a74a71a73a80a162a118 a9a153a101a17a71a91a62a63a80a53a72a54a71
a72a74a112a43a68a41a101a17a71a128a71a73a80a17a59a67a57a60a69a70a71a73a59a19a90a70a78a73a72a41a95a43a68a70a69a5a71a73a80a17a59a67a77a53a72a41a69a17a59a19a95a162a62a63a80a53a59a19a78a73a59a91a71a73a80a17a59a122a61a9a59a23a69a70a71a73a78a79a68a70a57a60a103a111a207a24a50a52a51a153a57a65a64a63a95a92a68a87a61a23a72a54a71a79a59a48a103 a27a110a191a117a80a17a59a25a4a15a78a73a59a19a59a23a69a162a188 a64
a96a114a101a17a69a53a61a9a71a79a57a60a68a41a69a102a112a17a95a60a68a54a62a76a64a63a101a17a77a50a72a54a71a76a71a73a80a17a59a185a61a9a59a23a69a75a71a79a78a73a68a41a57a65a103a34a118 a33a76a68a54a62a128a59a19a89a41a59a23a78a23a192a25a71a73a80a17a59a67a96a114a101a17a69a40a61a177a71a79a57a60a68a41a69a179a57a60a64a91a57a92a69a75a71a79a59a23a90a41a78a81a72a74a112a17a95a60a59a67a112a43a59a23a61a19a72a41a101a53a64a79a59
a71a79a80a53a59a179a57a60a69a70a71a73a59a19a90a70a78a73a72a41a69a53a103a157a112a17a95a60a68a54a62a76a64a85a101a17a77a199a72a74a71a12a72a45a78a81a72a54a71a73a59a36a71a73a80a53a72a54a71a12a57a60a64a12a64a66a95a60a68a54a62a128a59a19a78a107a71a73a80a53a72a74a69a84a71a73a80a17a59a179a78a81a72a54a71a73a59a179a72a54a71a12a62a63a80a17a57a65a61a81a80a84a71a79a80a17a59
a64a79a101a17a78a66a96a32a72a70a61a9a59a149a98a180a59a48a72a41a64a79a101a17a78a73a59a91a90a41a68a25a59a48a64a94a71a73a68a181a183a23a59a19a78a73a68a185a57a60a69a5a71a73a80a17a59a135a89a25a57a60a61a19a57a92a69a17a57a92a71a113a97a107a68a74a96a47a71a79a80a17a59a67a64a79a57a92a69a17a90a70a101a17a95a65a72a74a78a73a57a206a71a113a97a70a118 a115 a68a180a62a128a59a149a197a25a69a17a68a54a62a84a71a73a80a53a72a54a71
a210 a35
a71a79a80a53a59a122a57a92a69a75a71a73a59a19a90a41a78a81a72a74a95a34a59a19a209a87a57a60a64a66a71a73a64a76a72a41a69a53a103a12a57a60a64a63a208a53a69a17a57a92a71a79a59a41a192a40a112a17a101a87a71a91a57a65a64 a4a135a72a41a101a53a64a73a64a63a116a70a101a40a72a41a103a87a78a81a72a54a71a73a101a17a78a79a59a67a61a23a72a74a77a53a72a41a112a17a95a92a59a67a68a41a96a30a77a43a59a19a78a79a96a114a68a41a78a73a98a99a57a92a69a53a90
a62a128a59a19a95a60a95a34a57a92a69a12a71a79a80a53a59a67a77a17a78a79a59a48a64a66a59a23a69a53a61a9a59a135a68a41a96a47a71a79a80a53a57a60a64a76a64a79a57a60a69a17a90a41a101a17a95a65a72a74a78a73a57a92a71a113a97 a27
a10a76a138a51a137 a0 a12 a1 a1a15a5a34a151a87a158a53a144a1a0 a5 a9 a139 a12 a16a19a18a117a141a10a1 a156a4a3 a5
a2a7a161a65a160a162a161a32a163 a13 a13a162a14a17a42a45a44a46a55a32a22
a165a43a166a75a167a169a168a47a170a16a15 a4
a5
a29
a11
a29
a29
a2 a3
a60
a3
a38a41a207a61a20 a54 a56
a202a58
a29
a20
a202
a29
a2 a3
a60
a51
a3
The Singular
Kernel Problem
Symmetrized 1-D Exmaple
Note no 0ix =1
x
Quad Point
x
a172a174a173a34a175a177a176a18a17 a26
a104a105a69 a210 a190 a62a128a59a67a95a60a68a75a68a70a197a111a72a74a71a91a72a180a96a114a101a17a69a53a61a177a71a73a57a92a68a70a69
a18
a21a114a207 a23 a25
a29
a2 a3
a60
a3
a62a63a80a17a57a60a61a81a80a110a57a65a64a63a57a60a69a70a71a73a59a19a90a70a78a73a72a41a112a17a95a92a59a135a68a70a69a103a44a63a2a47a98a33a48a47a6a49a162a112a17a101a17a71a91a80a53a72a41a64
a72a99a64a66a57a60a69a17a90a70a101a17a95a60a72a41a78a79a57a92a71a113a97a85a72a54a71a63a207 a25a32a45a17a118
a2a7a161a65a160a162a161a65a160 a2 a28a52a20a48a22a25a49a53a39a54a14 a8a46a55a32a22a36a14a53a28a7a86 a15 a37a43a28a30a29a32a28a52a20a23a22a25a49a40a39a54a14 a8a30a55a51a22 a2a162a29a51a28a7a49a43a33a7a55a51a14a87a39a54a29a114a20a48a29a32a22a25a16
a165a43a166a75a167a169a168a47a170a20a19 a31
∞→ Areafinite Area →
x
1f(x) =
x
1f(x) =
a10a76a138a18a13 a4a50a152 a1 a156a149a141a30a158a40a144a66a148a6a5 a7 a147a76a141a10a9a76a158a53a141a7a151a17a147a91a158 a5 a0a117a150a53a159a13a5a14a1a15a5a47a142
a165a43a166a75a167a169a168a47a170a20a19a34a171
Gauss-Quad
Simple Quad
Number of Points
E
r
r
o
r
Approximately
Integrating
1
1
1 dx
x?"
Large errors
even with many
points!
a210 a32
a172a174a173a34a175a177a176a18a17 a11
a186a174a59a76a68a41a112a53a64a79a59a19a78a73a89a41a59a153a71a79a80a40a72a54a71 a4a135a72a41a101a53a64a79a64a94a116a70a101a40a72a41a103a87a78a81a72a54a71a73a101a17a78a79a59a117a57a65a64a143a69a17a68a41a71a58a89a41a59a19a78a73a97a185a90a70a68a25a68a25a103a180a72a54a71a153a57a92a69a75a71a79a59a23a90a41a78a81a72a54a71a73a57a92a69a17a90a15a71a73a80a17a57a60a64a58a96a114a101a17a69a53a61a9a71a79a57a60a68a41a69a162a118
a191a117a80a17a59a135a61a9a68a70a69a75a89a70a59a19a78a73a90a41a59a23a69a53a61a9a59a149a57a60a64a128a78a81a72a54a71a79a80a53a59a19a78a117a77a40a68a25a68a70a78a23a118 a109 a64a117a72a180a98a107a72a54a71a79a71a79a59a19a78a117a68a41a96a47a96a32a72a41a61a9a71a23a192a87a57a92a71a63a57a60a64a117a98a99a68a41a78a73a59a91a57a60a69a53a72a70a61a19a61a9a101a53a78a73a72a74a71a79a59a149a71a79a80a53a72a41a69
a71a79a80a53a59a180a64a79a57a92a98a99a77a17a95a60a59a107a116a75a101a53a72a41a103a87a78a81a72a54a71a73a101a17a78a73a59a181a64a73a61a81a80a17a59a19a98a99a59a41a118a67a104a105a69a50a71a79a80a53a59a181a69a17a59a19a209a25a71a15a96a114a59a23a62 a64a79a95a92a57a65a103a87a59a23a64a15a62a128a59a181a77a53a78a79a59a48a64a66a59a23a69a70a71a67a64a79a59a19a89a41a59a23a78a73a72a41a95a162a71a73a59a23a61a81a80a87a195
a69a17a57a65a116a75a101a17a59a23a64a117a68a41a96a30a80a53a72a41a69a53a103a87a95a60a57a92a69a53a90a180a57a60a69a75a71a79a59a23a90a41a78a81a72a74a95a65a64a153a62a63a57a92a71a79a80a102a64a79a57a92a69a53a90a41a101a17a95a65a72a74a78a73a57a206a71a73a57a92a59a48a64 a101a114a62a63a80a17a57a65a61a81a80a102a72a74a78a73a59a15a57a60a69a75a71a79a59a23a90a41a78a81a72a74a112a17a95a60a59a41a192a25a68a74a96a46a61a9a68a70a101a17a78a73a64a79a59 a102
a0
a115 a101a17a112a17a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95 a101a51a72a41a103a17a72a41a77a87a71a79a57a60a89a41a59 a102a153a116a75a101a53a72a70a103a87a78a81a72a54a71a79a101a53a78a79a59
a0
a88 a80a40a72a74a69a17a90a70a59a15a68a41a96a30a89a54a72a41a78a79a57a65a72a74a112a17a95a60a59a23a64a128a68a41a96a7a57a60a69a70a71a73a59a19a90a70a78a73a72a74a71a79a57a60a68a41a69
a0
a115 a57a92a69a17a90a70a101a17a95a65a72a74a78 a101a4a135a72a41a101a53a64a73a64a66a57a65a72a74a69a96a102a128a116a75a101a53a72a70a103a87a78a73a72a74a71a79a101a17a78a73a59
a10a76a138a11a10 a0 a1 a156a149a158a17a152a2a1 a5a14a9a4a3 a5a7a150a53a159a91a148a149a144a6a5a117a147 a5a7a142
a2a7a161a3a2a7a161a32a163 a2a47a33 a8a46a29a32a28a52a20a48a22a75a39a28a3a17a14a53a55a8a7 a5a99a86a7a14a87a44a30a20a23a29a30a3a40a22a10a9a107a83a107a33a30a14a17a86a7a39a54a14a87a20a23a33a7a39a54a22
a165a43a166a75a167a169a168a47a170a20a19 a15
SMA-HPC ?1999 MIT
Improved Techniques
x
1
x
a115 a101a17a112a52a103a87a57a60a89a25a57a60a103a87a59a135a71a73a80a17a59a67a57a60a69a70a71a73a59a19a90a70a78a73a72a74a71a79a57a60a68a41a69a5a57a60a69a75a71a79a59a23a78a79a89a54a72a74a95
a5
a29
a11
a29
a29
a2 a3
a60
a3
a38a41a207 a25
a5
a11
a30a13a12
a29
a11
a29
a29
a2 a3
a60
a3
a38a70a207a41a44
a5
a30
a11
a30 a12
a29
a29
a2 a3
a60
a3
a38a41a207a10a44
a5
a30a13a12
a29a30
a29
a2 a3
a60
a3
a38a70a207a41a44
a5
a29a30a13a12
a29
a29
a2 a3
a60
a3
a38a70a207 a4a91a64a79a59 a4a135a72a41a101a53a64a79a64
a116a75a101a53a72a41a103a17a78a73a72a74a71a79a101a17a78a73a59a149a57a60a69a110a59a23a72a70a61a81a80a111a64a79a101a17a112a17a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95
a56a46a68a41a95a60a97a25a69a17a68a41a98a99a57a65a72a74a95a65a64a94a208a53a71a91a64a66a101a17a112a53a57a92a69a75a71a79a59a23a78a79a89a54a72a41a95a60a64a117a112a43a59a9a71a79a71a79a59a19a78
a93a143a209a87a77a43a59a19a69a53a64a79a57a92a89a70a59a135a57a206a96a46a98a107a72a74a69a25a97a5a64a79a101a17a112a17a57a60a69a70a71a73a59a19a78a73a89a54a72a74a95a65a64a117a101a53a64a66a59a48a103a34a118
a2a7a161a3a2a7a161a65a160 a164 a26a30a14a17a28a7a49a43a22a36a37a43a35a12a11a67a14a87a39a54a29a32a14 a8a46a55a32a22a25a16 a31a122a35a38a37a53a39 a2a47a29a32a42a146a44a30a55a51a22 a164 a14a17a16a48a22a25a16
a165a43a166a75a167a169a168a47a170a20a19a46a19
a88 a80a53a72a41a69a17a90a41a59a15a89a54a72a74a78a73a57a60a72a41a112a17a95a92a59a48a64a128a71a79a68a99a59a19a95a60a57a60a98a180a57a60a69a53a72a74a71a79a59a122a64a79a57a92a69a17a90a70a101a17a95a65a72a74a78a73a57a206a71a113a97
a5
a42
a25a157a207
a7 a23 a5 a38 a5a100a25a57a38a70a207
a5
a29
a11
a29
a29
a2 a3
a60
a3
a38a41a207 a25 a23
a5
a29a30
a29
a2 a3a13
a14
a3
a23 a5 a38 a5a100a25 a23
a5
a29a30
a23 a38a26a5
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| 课件名称： | 麻省理工学院：偏微分方程式数字方法（英文版） |
| 课件分类： | 航空航天 |
| 课件类型： | 教学课件 |
| 文件大小： | 17.8MB |
| 下载次数： | 2 |
| 评论次数： | 2 |
| 用户评分： | 6.5 |
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