quadrature was invented two centuries ago,but adaptive quadrature didn't arrive until the 1960s.Adaptive ODE sovers came soon after,and turned the solution of most ordi- nary differential equations into the use of a black box.Partial differential equations are not yet boxed in black,but the trend is in that direction.As time goes by,adaptivity managed by the computer's intelligence becomes more and more widespread.Computers are not as wise as people.but they can explore a forest of possibilities faster than we can.In fifty years,this is how most numerical problems will be solved.We will tell the machine what we want,and the machine,an intelligent control system sitting atop an ency copaedia of numerical methods,will juggle computational options at inco mprehen- sible speed until it has solved the problem to the accuracy required.Then it will give us the answer;and if we insist,it may even tell us something of how it got there. The power unleashed by this kind of computing will be vast.Large parts of physical reality will be simulated in real time before our eyes,with effects so far beyond what the men of 1950couldenvision that the word"computation"may begin to seem old-fashione and drop out of use When computations are all intelligent,when everything is embedded in a contro loop.the mathematical landscape will change.One distinction that means a great deal to us today is that,broadly speaking,linear problems can be solved in one pass,but nonline ar ones require iter ation.In fifty years,when everything is embedded in an iterative loop anyway.this difference will have diminished.For the same reason,todav's big distinction between forward and inverse problems will have faded too My next prediction is a corollary Determinism in numerical computing will be gone Recently our family rented a car for a holiday.One evening we wanted to look at the stars,which meant turning off the dome light.We couldn't figure out how to do it! A decade ago,dlosing the doors and fipping a switch would have sfficed,bu it nowadays cars are more intelligent.In some,the light stays on for a fixed period after you dose the doors,and in ours,the situation was eve n more complicated.There was an interlock with the engine.plus some additional intelligence that we never got to the bottom of. Eventually we got the light off,but we were not quite sure how we had done it,or if we oould do it the same way again Have you noticed how many of our machines behave this way?photocopiers used to be deterministic,but nowadays they have complicated arrays of internal states.The first copy may come out in landscape orientation,but the second in portr ait,if the machine decides in-between that it ought to change modes.Typewriters used to be predictable too:you knew what would happen when you pressed a key.Nowadays,in Word or LaTeX,changing one char acter of input may alter the whole document in startling ways. Why,at motorway rest stops,even toilets are intelligent devices now whose states of mind we don't fully understand.and when you're finished with the toilet.you have two further negotiations to undertake with the intelligent sink and the intelligent hand drier! What's true of toilets will be true of numerical comput ations.In fifty vears,though the answers you get will be accurate without fail to the prescribed precision,you willquadrature was invented two centuries ago but adaptive quadrature didn t arrive until the s Adaptive ODE solvers came soon after and turned the solution of most ordi nary dierential equations into the use of a black box Partial dierential equations are not yet boxed in black but the trend is in that direction As time goes by adaptivity managed by the computer s intelligence becomes more and more widespread Computers are not as wise as people but they can explore a forest of possibilities faster than we can In fty years this is how most numerical problems will be solved We will tell the machine what we want and the machine an intelligent control system sitting atop an encyclopaedia of numerical methods will juggle computational options at incomprehen sible speed until it has solved the problem to the accuracy required Then it will give us the answer and if we insist it may even tell us something of how it got there The power unleashed by this kind of computing will be vast Large parts of physical reality will be simulated in real time before our eyes with eects so far beyond what the men of  could envision that the word computation may begin to seem old fashioned and drop out of use When computations are all intelligent when everything is embedded in a control loop the mathematical landscape will change One distinction that means a great deal to us today is that broadly speaking linear problems can be solved in one pass but nonlinear ones require iteration In fty years when everything is embedded in an iterative loop anyway this dierence will have diminished For the same reason today s big distinction between forward and inverse problems will have faded too My next prediction is a corollary  Determinism in numerical computing wil l be gone Recently our family rented a car for a holiday One evening we wanted to look at the stars which meant turning o the dome light We couldn t gure out how to do it A decade ago closing the doors and ipping a switch would have suced but nowadays cars are more intelligent In some the light stays on for a xed period after you close the doors and in ours the situation was even more complicated There was an interlock with the engine plus some additional intelligence that we never got to the bottom of Eventually we got the light o but we were not quite sure how we had done it or if we could do it the same way again Have you noticed how many of our machines behave this way Photocopiers used to be deterministic but nowadays they have complicated arrays of internal states The rst copy may come out in landscape orientation but the second in portrait if the machine decides in between that it ought to change modes Typewriters used to be predictable too you knew what would happen when you pressed a key Nowadays in Word or LaTeX changing one character of input may alter the whole document in startling ways Why at motorway rest stops even toilets are intelligent devices now whose states of mind we don t fully understand and when you re nished with the toilet you have two further negotiations to undertake with the intelligent sink and the intelligent hand drier What s true of toilets will be true of numerical computations In fty years though the answers you get will be accurate without fail to the prescribed precision you will