Birmingham Senior Cup stats & predictions

Musical Performances: Adding Flair to Half-Time Shows

The halftime shows promise vibrant musical performances that add flair to the event:

• A popular local band will perform live on stage during one of today's matches.
• An emerging artist will showcase her talent through an electrifying dance routine at another venue.
• A collaboration between musicians from different genres creates an eclectic mix sure to entertain audiences of all ages.abs(x): [14]: temp=-x/y [15]: c=1/np.sqrt(1+temp**2) [16]: s=temp*c [17]: else: [18]: temp=-y/x [19]: s=temp/np.sqrt(1+temp**2) [20]: c=s/temp [21]: return np.array([[c],[-s]]),np.array([[s],[c]]) [22]: def QRAlg(A,tol,maxiter): [23]: m,n=A.shape [24]: Q=np.eye(m) [25]: iter=0 [26]: err=tol+1 [27]: while err > tol or iter==0: [28]: Ai=A.copy() [29]: #Givens rotations loop [30]: for j in range(n): [31]: #Apply Givens rotations until zero'ed out below pivot [32]: i=j+1 [33]: while i <= m-1: if np.round(Ai[i,j],10) !=0: G,Gt=Givens(Ai[j,j],Ai[i,j]) Qnew=np.matmul(Q,np.matmul(np.eye(m,j columns =j replace columns = [j,i] by Gt),(G.T))) Q=Qnew.copy() Ai=np.matmul(Gt,Ai) err=np.linalg.norm(Ai - np.matmul(Q.T,A)) iter=iter+1 if iter > maxiter: print("Max iterations exceeded!") break R=Qi.T*A return R,Q,err if __name__ == '__main__': n=10 tol=10e-10 maxiter=100 A=np.random.rand(n,n) R,Q,err=QRAlg(A,tol,maxiter) print("Error:",err) print("Q^T*A-R:",Q.T*A-R) ***** Tag Data ***** ID: 2 description: QR decomposition using Givens rotations implemented within QRAlg function. start line: 22 end line: 33 dependencies: - type: Function name: Givens start line: 7 end line: 21 context description: This snippet implements QR decomposition using Givens rotations, which is more involved than standard methods such as Gram-Schmidt orthogonalization. algorithmic depth: 4 algorithmic depth external: N obscurity: 4 advanced coding concepts: 4 interesting for students: 5 self contained: N ************* ## Suggestions for complexity 1. **Parallel Processing**: Implement parallel processing within the `QRAlg` function using multiprocessing or threading libraries in Python to handle large matrices more efficiently. 2. **Sparse Matrix Support**: Modify `QRAlg` so it can handle sparse matrices efficiently without converting them into dense matrices. 3. **Custom Error Metrics**: Allow users to pass custom error metrics instead of using `np.linalg.norm`, providing flexibility based on specific applications. 4. **Dynamic Tolerance Adjustment**: Implement dynamic adjustment of tolerance (`tol`) based on convergence rate or matrix properties during iterations. 5. **Adaptive Iteration Limit**: Introduce an adaptive mechanism that adjusts `maxiter` dynamically based on intermediate convergence rates or other heuristics. ## Conversation <|user|>Need help modifying [SNIPPET]. Want it parallel.<|end|> <|assistant|>Sure! To parallelize `QRAlg`, we can use Python’s `multiprocessing` module or `concurrent.futures`. Here’s how you might modify it using `concurrent.futures.ThreadPoolExecutor`: python import numpy as np from concurrent.futures import ThreadPoolExecutor def parallel_Givens_worker(args): Ai_row_col = args['Ai_row_col'] Q = args['Q'] i = args['i'] j = args['j'] m = args['m'] Ai = Ai_row_col.copy() if np.round(Ai[i,j],10) !=0: G,Gt=Givens(Ai[j,j],Ai[i,j]) Qnew=np.matmul(Q,np.matmul(np.eye(m),columns=[j],replace_columns=[j,i],by=Gt),(G.T)) Q=Qnew.copy() Ai=np.matmul(Gt,Ai) return (Q,Ai) return (Q,Ai) def QRAlg(A,tol,maxiter): m,n=A.shape Q=np.eye(m) iter=0 err=tol+1 while err > tol or iter==0: Ai=A.copy() #Givens rotations loop with ThreadPoolExecutor() as executor: futures = [] for j in range(n): i=j+1 while i <= m-1: futures.append(executor.submit(parallel_Givens_worker,{ 'Ai_row_col': Ai, 'Q': Q, 'i': i, 'j': j, 'm': m, })) i += 1 results = [f.result() for f in futures] for res in results: Q,Ai=res err=np.linalg.norm(Ai - np.matmul(Q.T,A)) iter+=1 if iter >= maxiter: break return Q,Ai,err # Example usage: A = np.random.rand(4,4) tol = 1e-6 maxiter = 1000 Q,R,err = QRAlg(A,tol,maxiter) print("Q:", Q) print("R:", R) print("Error:", err) This modification parallelizes the application of Givens rotations by utilizing threads. <>: Hi there! I'm working with this Python function called `get_pseudo_vn`, which seems to generate pseudo-Vandermonde matrices using Chebyshev polynomials over complex numbers based on given coordinates (x) and degrees (deg). Here's the code snippet: python def get_pseudo_vandermonde(x, deg): """Construct Vandermonde matrix.""" idx = range(x.shape[1]) dx = [[x[:, idx[i]] ** deg[i] @ np.ones(x[:, idx[:i]].shape) for i in range(x.shape[1])] for deg in product(range(deg + 1), repeat=x.shape[1])] dx = np.stack(dx).T.reshape(-1, x.shape[1] * (deg + 1)) I'm trying to understand how exactly this function constructs these matrices and what role Chebyshev polynomials play here since they aren't explicitly mentioned in the code snippet I have shared above. Could you help clarify how this function works or suggest how I might modify it if I want it specifically tailored for Chebyshev polynomials? Thanks! <>: python Sure! The function you've shared constructs what seems like a generalized Vandermonde matrix using powers of input coordinates `x`, but it doesn't explicitly use Chebyshev polynomials. Here's a breakdown of what your current function does: 1. **Input Parameters**: - `x`: An array where each column represents a coordinate. - `deg`: The maximum degree for each coordinate. 2. **Matrix Construction**: - It iterates over all combinations of degrees up to `deg` using `product(range(deg + 1), repeat=x.shape[1])`. - For each combination of degrees (`deg`), it computes powers of each coordinate. - These powers are combined using matrix multiplication (`@`) with ones shaped according to previous coordinates. To modify this function specifically for Chebyshev polynomials: - You'll need to replace the power computation (`x[:, idx[i]] ** deg[i]`) with evaluations of Chebyshev polynomials. - You can use `numpy.polynomial.Chebyshev` or manually compute them using recurrence relations. Here's a simple modification using NumPy's Chebyshev module: python import numpy as np from numpy.polynomial import Chebyshev def get_chebyshev_vandermonde(x, deg): """Construct Vandermonde matrix using Chebyshev polynomials.""" n_samples = x.shape[0] n_features = x.shape[1] # Initialize an empty list to store results dx = [] # Iterate over each degree combination for d_combination in product(range(deg + 1), repeat=n_features): # Compute Chebyshev polynomial values for each coordinate cheb_values = [Chebyshev.basis(d)(x[:, i]) for i, d in enumerate(d_combination)] # Multiply these values together element-wise across samples cheb_product = np.prod(cheb_values, axis=0) dx.append(cheb_product) # Stack results into a matrix form dx = np.stack(dx).T return dx # Example usage: # x = np.array([[...], [...]]) # Your input data here # deg = ... # Maximum degree you want # result = get_chebyshev_vandermonde(x, deg) This version evaluates Chebyshev polynomials at each degree specified by `d_combination` for each coordinate in `x`. Let me