![]() ![]() So the vectorized way to describeĪ quadratic form like this is to take a matrix, a two by two matrix since You would just change what b represents but you'll see why it's moreĬonvenient to write it this way in just a moment. Thinking of b times xy, we actually think of thisĪs two times some constant times xy and this of courseĭoesn't make a difference. Of show you how we do it and then we'll work it through Start introducing things like a hundred variables, it would get seriously out of hand because there's a lot ofĭifferent quadratic terms so we want a nice way to express this. ![]() The xz quadratic term and then some other constant times the z squared quadratic term and another one for the yz quadratic term and it would get out of hand and as soon as you Have some other term, some other constant times Introducing the variable z then you would have to So the question is can be we do something similar like that with our quadratic form? Because you can imagine let's say we started Hundred corresponding variables and the notation doesn'tīecome any more complicated. Writing things down like this is that v could be a vector that contains not just three numbers but a hundred numbers and then x would have a Number times a variable number, it's kind of like taking a constant vector times a variable vector. Just kind of looks like a constant times a variable just like in the single variable world when you have a constant Whole constant vector and then you can write down, take the dot product between that and then have another symbol, maybe a bold faced x which represents a vector thatĬontains all of the variables and this way, your notation Then you can have just a symbol like a v let's say which represents this Like that to each other, we can express this nicely with vectors where you pile all of theĬonstants into their own vector, a vector containing a, b and c and you imagine the dot product between that and a vector that contains all of the variable components, x, y and z and the convenience here is If you see something like this where every variable is justīeing multiplied by a constant and then you add terms Think about linear terms where let's say you have a times x plus b times y and I'll throw another variable in there, another constant times another variable z. In a vectorized sense? And for analogy, let's Intimidating than it needs to be but anyways, so we have a quadratic form and the question is how can we express this Purely quadratic terms but of course, mathematiciansĭon't want to call it just a purely quadratic expression instead they have to giveĪ fancy name to things so that it seems more It's not the case that you have an x term sitting on its own or a constant out here like two when you're addingĪll of those together instead it's just you have That the only things in here are quadratic. Means something is squared or you have two variables but why do they call it a form? And basically it just means I always kind of was like, what, what does form mean? I know what a quadratic expression is and quadratic typically Now, this kind ofĮxpression has a fancy name. Some kind of expression that looks like a times x squared and I'm thinking x is a variable times b times xy, y is another variable, plus c times y squared and I'm thinking of a, bĪnd c as being constants and x and y as being variables. Of multivariable functions which is a mouthful to say so let's say you have I need to talk about before I can describe the vectorized form for the quadratic approximation Copy down and over until my columns are just copying zeros.- Hey guys. I copy just my Z column and paste in a new column (say starting in Cell F1 down to Cell F1764). Then I do custom sort, first by X2, then by Y2 so my data reads (1, 1, Z) (1, 2, Z). I sort my data by X (data), and fill X2 column 1, 1, 1., 2, 2, 2. I created three new columns: (#, X2 and Y2): # just lists integers 1 to 1764 (42x42 array). Why not force them into integers, and then sort. The values hover around 42 discrete values. My X and Y locations don't line up perfectly: 1.001, 0.998, 1.004., but these are just addresses on my grid. I did find an easier work-around versus my original brute-force method. Hi Fluff, I could not get your suggestions to work. ![]()
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