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Compute the gradients and Hessians for the following functions. You are encouraged to use the chain rule as needed. You aren't required to open up the indices or compute element-wise for every problem, although for some problems it may be useful to do so. i. 9₁(x) = ™ Az ii. 92(x) = || A - 6||² iii. 93(2) = log (et) iv. (Practice) 94(7) = log (₁-1 eªi ¹2-bi) v. (Practice) 95 (7) = ellAz-b||² Consider the case now where all vectors and matrices above are scalar; do your answers above make sense? (No need to answer this in your submission)

Compute the gradients and Hessians for the following functions. You are encouraged to use the chain rule as needed. You aren't required to open up the indices or compute element-wise for every problem, although for some problems it may be useful to do so. i. 9₁(x) = ™ Az ii. 92(x) = || A - 6||² iii. 93(2) = log (et) iv. (Practice) 94(7) = log (₁-1 eªi ¹2-bi) v. (Practice) 95 (7) = ellAz-b||² Consider the case now where all vectors and matrices above are scalar; do your answers above make sense? (No need to answer this in your submission)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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4. Gradients, Jacobian matrices and Hessians The Gradient of a scalar-valued function g : R → R, is the column vector of length n, denoted as Vg, containing the derivatives of components of g with respect to the variables: (Vg(x))i The Hessian of a scalar-valued function g : R" → R, is the n x n matrix, denoted as V²g, containing the second derivatives of components of g with respect to the variables: (V²g(x))ij = i. 9₁(x) = 7¹ Az ii. 92(x) = || A iii. 93(7) = log The Jacobian of a vector-valued function g: R" → Rm is the m x n matrix, denoted as Dg, containing the derivatives of components of g with respect to the variables: (Dg)ij = - 6||²/2 Σei dg əxi 0²g -(x), i=1,...,n, j = 1,...,n. дх;дх, Əgi əxj ·(x), i = 1,...n. " For the remainder of the class, we will repeatedly have to take Gradients, Hessians and Jacobians of functions we are trying to optimize. This exercise serves as a warm up for future problems. i. g(x₁, x₂) = ii. g(x₁, x2) = x1x2 (a) Compute the gradients and Hessians for the following functions. You are encouraged to use the chain rule as needed. You aren't required to open up the indices or compute element-wise for every problem, although for some problems it may be useful to do so. i= 1,...,m, j = 1,..., n. iv. (Practice) 94(7) = log (₁1 e¹-b₁ v. (Practice) 95(x) = e||A-b||²| Consider the case now where all vectors and matrices above are scalar; do your answers above make sense? (No need to answer this in your submission) (b) Compute the Jacobians for the following maps i. g(x) = Az ii. g(x) = f(x) where f: R" → R is once-differentiable iii. (Practice) g(x) = f(A + b) where f: R" → R is once differentiable and A € Rnxn (c) Plot/hand-draw the level sets of the following functions: x²x² + 4 9 Also point out the gradient directions in the level-set diagram. Additionally, compute the first and second order Taylor series approximation around the point (1, 1) for each function and comment on how accurately they approximate the true function.
Transcribed Image Text:4. Gradients, Jacobian matrices and Hessians The Gradient of a scalar-valued function g : R → R, is the column vector of length n, denoted as Vg, containing the derivatives of components of g with respect to the variables: (Vg(x))i The Hessian of a scalar-valued function g : R" → R, is the n x n matrix, denoted as V²g, containing the second derivatives of components of g with respect to the variables: (V²g(x))ij = i. 9₁(x) = 7¹ Az ii. 92(x) = || A iii. 93(7) = log The Jacobian of a vector-valued function g: R" → Rm is the m x n matrix, denoted as Dg, containing the derivatives of components of g with respect to the variables: (Dg)ij = - 6||²/2 Σei dg əxi 0²g -(x), i=1,...,n, j = 1,...,n. дх;дх, Əgi əxj ·(x), i = 1,...n. " For the remainder of the class, we will repeatedly have to take Gradients, Hessians and Jacobians of functions we are trying to optimize. This exercise serves as a warm up for future problems. i. g(x₁, x₂) = ii. g(x₁, x2) = x1x2 (a) Compute the gradients and Hessians for the following functions. You are encouraged to use the chain rule as needed. You aren't required to open up the indices or compute element-wise for every problem, although for some problems it may be useful to do so. i= 1,...,m, j = 1,..., n. iv. (Practice) 94(7) = log (₁1 e¹-b₁ v. (Practice) 95(x) = e||A-b||²| Consider the case now where all vectors and matrices above are scalar; do your answers above make sense? (No need to answer this in your submission) (b) Compute the Jacobians for the following maps i. g(x) = Az ii. g(x) = f(x) where f: R" → R is once-differentiable iii. (Practice) g(x) = f(A + b) where f: R" → R is once differentiable and A € Rnxn (c) Plot/hand-draw the level sets of the following functions: x²x² + 4 9 Also point out the gradient directions in the level-set diagram. Additionally, compute the first and second order Taylor series approximation around the point (1, 1) for each function and comment on how accurately they approximate the true function.
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