<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class="">Even with cancellation, the number of bits required to represent the exact terms involved in computing the inverse grow like O(n log n) in the typical case, IIRC. So exact arithmetic rapidly becomes unfeasible for even modestly large matrices.</div><div class=""><br class=""></div><div class="">Aside from purely symbolic matrices that come up in theoretical physics and matrices over finite fields, the entries of a matrix are rarely known exactly, so the rich set of approximate-rank-revealing and numerically stable methods that we have for floating-point turn out to be much more useful anyway.</div><div class=""><br class=""></div><div class="">That’s not to say that it isn’t fun to play around with these concepts, of course =)</div><div class=""><br class=""></div><div class="">– Steve</div><br class=""><div><blockquote type="cite" class=""><div class="">On Jan 13, 2016, at 11:27 PM, Matt Whiteside <<a href="mailto:mwhiteside.dev@gmail.com" class="">mwhiteside.dev@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><meta http-equiv="Content-Type" content="text/html charset=utf-8" class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><br class=""><div class=""><blockquote type="cite" class=""><div class="">On Jan 13, 2016, at 11:16, Stephen Canon <<a href="mailto:scanon@apple.com" class="">scanon@apple.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><meta http-equiv="Content-Type" content="text/html charset=utf-8" class=""><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class="">… in exact arithmetic, not in floating-point, which is what people tend to actually use for matrices. For that matter, the entire notion of “invertible” as a binary attribute is problematic when dealing with approximate arithmetic; a matrix may be formally invertible but so ill-conditioned that the result is effectively meaningless.</div><div class=""><br class=""></div><div class="">Not really trying to shoot you down, just observing that this is an extremely subtle example with lots of hidden sharp edges.</div><div class=""><br class=""></div><div class="">– Steve</div></div></div></blockquote></div><br class=""></div><div class="">That’s a good point. So then I wonder if exact arithmetic wouldn’t be good enough for a lot of entry level math & science applications, especially the ones where this type of approach would come to mind in the first place.</div><div class=""><br class=""></div><div class="">Something like,</div><div class=""><br class=""></div><blockquote style="margin: 0 0 0 40px; border: none; padding: 0px;" class=""><div style="margin: 0px; line-height: normal; font-family: Menlo;" class=""><div style="margin: 0px; line-height: normal; color: rgb(112, 61, 170);" class=""><span style="font-variant-ligatures: no-common-ligatures; color: rgb(187, 44, 162);" class="">struct</span><span style="font-variant-ligatures: no-common-ligatures;" class=""> Rational:</span><span style="font-variant-ligatures: no-common-ligatures;" class="">CustomDebugStringConvertible</span><span style="font-variant-ligatures: no-common-ligatures;" class="">{</span></div></div></blockquote><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class=""><span class="Apple-tab-span" style="white-space:pre"> </span></span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(187, 44, 162);" class="">let</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class=""> numerator, denominator: </span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(112, 61, 170);" class="">Int</span><br class=""><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class=""><span class="Apple-tab-span" style="white-space:pre"> </span></span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(187, 44, 162);" class="">var</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class=""> debugDescription:</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(112, 61, 170);" class="">String</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class="">{</span><br class=""><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class=""><span class="Apple-tab-span" style="white-space:pre"> </span> </span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(187, 44, 162);" class="">get</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class="">{</span><br class=""><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class=""><span class="Apple-tab-span" style="white-space:pre"> </span> </span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(187, 44, 162);" class="">return</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class=""> </span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(209, 47, 27);" class="">"</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class="">\</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(209, 47, 27);" class="">(</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(112, 61, 170);" class="">Double</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class="">(</span><span style="color: rgb(79, 129, 135); font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class="">numerator</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class="">)/</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(112, 61, 170);" class="">Double</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class="">(</span><span style="color: rgb(79, 129, 135); font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class="">denominator</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures;" class="">)</span><span style="font-family: Menlo; font-variant-ligatures: no-common-ligatures; color: rgb(209, 47, 27);" class="">)"</span><br class=""><span class="Apple-tab-span" style="font-family: Menlo; white-space: pre;"> </span> <span style="font-family: Menlo;" class="">}</span><br class=""><span class="Apple-tab-span" style="font-family: Menlo; white-space: pre;"> </span><span style="font-family: Menlo;" class="">}</span><br class=""><blockquote style="margin: 0 0 0 40px; border: none; padding: 0px;" class=""><div style="margin: 0px; line-height: normal; font-family: Menlo;" class=""><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures" class="">}</span></div><div style="margin: 0px; line-height: normal; min-height: 21px;" class=""><span style="font-variant-ligatures: no-common-ligatures" class=""></span><br class=""></div><div style="margin: 0px; line-height: normal; color: rgb(79, 129, 135);" class=""><span style="font-variant-ligatures: no-common-ligatures; color: #bb2ca2" class="">func</span><span style="font-variant-ligatures: no-common-ligatures;" class=""> *(lhs:</span><span style="font-variant-ligatures: no-common-ligatures" class="">Rational</span><span style="font-variant-ligatures: no-common-ligatures;" class="">,rhs:</span><span style="font-variant-ligatures: no-common-ligatures" class="">Rational</span><span style="font-variant-ligatures: no-common-ligatures;" class="">) -> </span><span style="font-variant-ligatures: no-common-ligatures" class="">Rational</span><span style="font-variant-ligatures: no-common-ligatures;" class="">{</span></div><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures;" class=""> </span><span style="font-variant-ligatures: no-common-ligatures; color: #bb2ca2" class="">return</span><span style="font-variant-ligatures: no-common-ligatures" class=""> </span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">Rational</span><span style="font-variant-ligatures: no-common-ligatures" class="">(numerator: lhs.</span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">numerator</span><span style="font-variant-ligatures: no-common-ligatures" class="">*rhs.</span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">numerator</span><span style="font-variant-ligatures: no-common-ligatures" class="">, </span></div><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures" class=""> denominator: lhs.</span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">denominator</span><span style="font-variant-ligatures: no-common-ligatures" class=""> * rhs.</span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">denominator</span><span style="font-variant-ligatures: no-common-ligatures" class="">)</span></div><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures" class="">}</span></div><div style="margin: 0px; line-height: normal; min-height: 21px;" class=""><span style="font-variant-ligatures: no-common-ligatures" class=""></span><br class=""></div><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures; color: #bb2ca2" class="">func</span><span style="font-variant-ligatures: no-common-ligatures" class=""> *(lhs:</span><span style="font-variant-ligatures: no-common-ligatures; color: #703daa" class="">Int</span><span style="font-variant-ligatures: no-common-ligatures" class="">,rhs:</span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">Rational</span><span style="font-variant-ligatures: no-common-ligatures" class="">) -> </span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">Rational</span><span style="font-variant-ligatures: no-common-ligatures" class="">{</span></div><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures;" class=""> </span><span style="font-variant-ligatures: no-common-ligatures; color: #bb2ca2" class="">return</span><span style="font-variant-ligatures: no-common-ligatures" class=""> </span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">Rational</span><span style="font-variant-ligatures: no-common-ligatures" class="">(numerator: lhs*rhs.</span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">numerator</span><span style="font-variant-ligatures: no-common-ligatures" class="">, </span></div><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures" class=""> denominator: rhs.</span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">denominator</span><span style="font-variant-ligatures: no-common-ligatures" class="">)</span></div><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures" class="">}</span></div><div style="margin: 0px; line-height: normal; min-height: 21px;" class=""><span style="font-variant-ligatures: no-common-ligatures" class=""></span><br class=""></div><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures; color: #bb2ca2" class="">let</span><span style="font-variant-ligatures: no-common-ligatures" class=""> π = </span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">Rational</span><span style="font-variant-ligatures: no-common-ligatures" class="">(numerator: </span><span style="font-variant-ligatures: no-common-ligatures; color: #272ad8" class="">355</span><span style="font-variant-ligatures: no-common-ligatures" class="">, denominator: </span><span style="font-variant-ligatures: no-common-ligatures; color: #272ad8" class="">113</span><span style="font-variant-ligatures: no-common-ligatures" class="">)</span></div><div style="margin: 0px; line-height: normal;" class=""><span style="font-variant-ligatures: no-common-ligatures; color: #bb2ca2" class="">let</span><span style="font-variant-ligatures: no-common-ligatures" class=""> two_π = </span><span style="font-variant-ligatures: no-common-ligatures; color: #272ad8" class="">2</span><span style="font-variant-ligatures: no-common-ligatures" class=""> </span><span style="font-variant-ligatures: no-common-ligatures; color: #31595d" class="">*</span><span style="font-variant-ligatures: no-common-ligatures" class=""> </span><span style="font-variant-ligatures: no-common-ligatures; color: #4f8187" class="">π</span></div><div class=""><br class=""></div></div></blockquote><div class=""><div style="margin: 0px; line-height: normal; min-height: 21px;" class=""><br class=""></div><div style="margin: 0px; line-height: normal; min-height: 21px;" class="">might get you pretty far with 64 bits.</div><div style="margin: 0px; line-height: normal; min-height: 21px;" class=""><br class=""></div><div style="margin: 0px; line-height: normal; min-height: 21px;" class="">Matt</div></div></div></div></blockquote></div></body></html>