Implementation and performance evaluation of iterative solver for multiple linear systems that have a common coefficient matrix
|
|
- Πανδώρα Δεσποίνη Παπακώστας
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Implementation and performance evaluation of iterative solver for multiple linear systems that have a common coefficient matrix, 1-1, almisofte@gmail.com, , keno@riken.jp, 1-1, yokokawa@port.kobe-u.ac.jp Seigo Imamura, Graduate School of System Informatics Kobe University, 1-1 Rokkodai-cho Nada-ku Kobe, Japan Kenji Ono, RIKEN AICS, Minatojima-minami-cho, Chuo-ku, Kobe, Japan Mitsuo Yokokawa, Graduate School of System Informatics Kobe University, 1-1 Rokkodai-cho Nada-ku Kobe, Japan Capacity computing is a promising scenario for improving performance on upcoming exascale supercomputers. Ensemble computing is an instance and has multiple linear systems associated with a common coefficient matrix. We implement to reduce load cost of coefficient matrix by solving them at the same time and improve performance of several iterative solvers. The maximum performance on SPARC64 XIfx on a node of FX1 was 6.8 times higher than that of naïve implementation. Finally, to deal with the different convergence processes of linear systems, we proposed control methods to skip the calculation of already converged vectors and conduct more acceleration. 1. CPU CPU 1 Capacity Computing (1) Capacity Computing CFD CFD Poisson CPU Byte/Flop B/F B/F Roofline model (2) Operational intensity Flop/Byte Operational intensity Compressed Row Storage CRS (3) Bit (4),(5) B/F 2. CFD ( p) = div( u t ) ϕ (1) p u ϕ (1) Cartesian (6) Bi-CGstab (7) Bi-CGstab SOR 3 4 Bi-CGstab 3. Fortran SOR List 1 Inner loop naïve List 2 Outer loop Lists 1,2 p, b, bp Ax = b x b A ac i, j, k l 1 do k=1, kx 2 do j=1, jx 3 do i=1, ix List 1: Pseudo code of SOR using Inner loop. 4 ndag_e = bp (1,i,j, k)! e 5 ndag_w = bp (2,i,j, k)! w 1 Copyright c 215 by JSFM
2 6 ndag_n = bp (3,i,j, k)! n 7 ndag_s = bp (4,i,j, k)! s 8 ndag_t = bp (5,i,j, k)! t 9 ndag_b = bp (6,i,j, k)! b 1 dd = bp (7,i,j, k)! diagonal 11 ac = bp (8,i,j, k)! active 12 dx = 1. / ac do l=1, lx 15 pp = p(l,i,j, k) 16 bb = b(l,i,j, k) ss = ndag_e *p(l,i+1,j,k )& 19 + ndag_w *p(l,i -1,j,k )& 2 + ndag_n *p(l,i,j+1,k )& 21 + ndag_s *p(l,i,j -1,k )& 22 + ndag_t * p(l, i,j,k +1)& 23 + ndag_b * p(l, i,j,k -1) 24 dp = ((ss -bb )*dx -pp )* omg 25 pn = pp + dp* ac 26 p(l,i,j, k) = pn de = dble (bb -(ss -pn*dd )) 29 res (l) = res (l) + de*de * ac 3 end do 31 end do 32 end do 33 end do 1 do l=1, lx 2 do k=1, kx 3 do j=1, jx 4 do i=1, ix List 2: Pseudo code of SOR using Outer loop. 5 ndag_e = bp (1,i,j, k)! e 6 ndag_w = bp (2,i,j, k)! w 7 ndag_n = bp (3,i,j, k)! n 8 ndag_s = bp (4,i,j, k)! s 9 ndag_t = bp (5,i,j, k)! t 1 ndag_b = bp (6,i,j, k)! b 11 dd = bp (7,i,j, k)! diagonal 12 ac = bp (8,i,j, k)! active pp = p(i,j,k, l) 15 bb = b(i,j,k, l) ss = ndag_e *p(i+1,j,k,l)& 18 + ndag_w *p(i -1,j,k,l)& 19 + ndag_n *p(i,j+1,k,l)& 2 + ndag_s *p(i,j -1,k,l)& 21 + ndag_t *p(i,j,k+1,l)& 22 + ndag_b *p(i,j,k -1,l) 23 dp = (( ss - bb )/dd -pp )* omg 24 pn = pp + dp* ac 25 p(i,j,k, l) = pn de = dble (bb -(ss -pn*dd )) 28 res (l) = res (l) + de*de * ac 29 end do 3 end do 31 end do 32 end do Operational intensity Flop/Byte SPARC64 IXfx 1 Outer loop dimension(ix, jx, kx, lx) Inner loop dimension(lx, ix, jx, kx) ix, jx, kx= 128 i Operational intesity flops Outer loop 3flops Inner loop lx Bytes Outer loop bp 1 8 p i+1, i, i-1, j+1, j-1, k+1, k-1 b, res p 5 1 Inner loop bp Outer loop p l, i+1, i, i-1, j+1, j-1, k+1, k-1 lx 4 k+1, k-1 2 b, res p res lx Tab Inner loop Outer loop 2.2 Inner loop k loop OpenMP Outer loop k loop Outer loop Red-Black SOR (8) R-B SOR Inner loop List 3 R-B SOR SOR i loop R-B SOR i loop Operational intensity Tab. 1 Tab. 1: Characteristics for two types of implementations. Grid Size = Outer Inner(RHS=lx < 4) Load & Store lx Arithmetic lx F/B Grid Size = Inner(RHS=lx > 4) Inner(RHS=128) Load & Store lx Arithmetic lx F/B List 3: Pseudo code of Red-Black SOR using Inner loop. 1 do color =,1 2 do k=1, kx 3 do j=1, jx 4 do i =1+ mod (k+j+ color,2),ix,2 5 ndag_e = bp (1,i,j, k)! e 6 ndag_w = bp (2,i,j, k)! w 7 ndag_n = bp (3,i,j, k)! n 8 ndag_s = bp (4,i,j, k)! s 9 ndag_t = bp (5,i,j, k)! t 1 SPARC64 IXfx 1flop 8flops 2 Copyright c 215 by JSFM
3 1 ndag_b = bp (6,i,j, k)! b 11 dd = bp (7,i,j, k)! diagonal 12 ac = bp (8,i,j, k)! active dx = 1. / dd do l=1, lx 17 pp = p(l,i,j, k) 18 bb = b(l,i,j, k) 19 2 ss = ndag_e * p(l, i+1, j,k )& 21 + ndag_w * p(l,i -1, j,k )& 22 + ndag_n * p(l, i,j+1, k )& 23 + ndag_s * p(l, i,j -1, k )& 24 + ndag_t * p(l, i,j,k +1)& 25 + ndag_b * p(l, i,j,k -1) 26 dp = ((ss -bb )*dx -pp )* omg 27 pn = pp + dp* ac 28 p(l,i,j, k) = pn 29 3 de = dble (bb -(ss -pn*dd )) 31 res (l) = res (l) + de*de * ac 32 end do end do 35 end do 36 end do 37 end do 3.1 Outer loop Inner loop Tab. 2 SPARC64 IXfx FX1 Tab. 3 (1) 3 2 ϕ = Dirichlet/Neumann Performance Monitor library PMlib (9) PMlib -Kfast,parallel,ocl,preex,array private,auto -Kopenmp,simd=2,uxsimd Tab. 2: Specification of a computing node of FX1. CPU SPARC64 XIfx Clock 1.65 GHz Core 16 Ideal peak performance Cache 12 MB Memory 32GB Bandwidth 85 GB/s Inner loop SOR R-B SOR 64 Outer loop R-B SOR 128 SOR Outer loop.6 Inner loop Operational intensity Inner loop SOR Outer loop Outer Inner Outer Inner Number of RHS vectors(l) (a) SOR method Number of RHS vectors(l) (b) R-B SOR method Fig. 1: Comparison of sequential performance with a problem size Tab. 3: Investigated parameters. Solver SOR method, Red Black SOR method Grid size 32 3, 64 3, 128 3, to 16 Number of pressure vectors 1, 2, 4, 8, 16, 32, 64, 128 SOR R-B SOR Fig. 1 Outer loop SOR R-B SOR Figs. 2, 3 SOR Outer Inner loop R-B SOR Outer loop Fig. 3(a) Inner loop Fig. 3(b) SOR Inner loop SOR % Outer loop SOR Copyright c 215 by JSFM
4 RHS=1 RHS=2 RHS=4 RHS=8 RHS=16 RHS=32 RHS=64 RHS= RHS=1 RHS=2 RHS=4 RHS=8 RHS=16 RHS=32 RHS=64 RHS= (a) Outer loop (a) Outer loop (b) Inner loop (b) Inner loop Fig. 2: Threading performance of SOR method with a problem size Fig. 3: Threading performance of R-B SOR method with a problem size Copyright c 215 by JSFM
5 4. Bi-CGstab Bi-CGstab Outer loop if l Inner loop if Inner loop if l if if 2 1 Fig. 4(a) l 1 Fig. 4(b) if Algorithm 1 Bi-CGstab x k, b, r, r k, p k, q Fig. 5 X, B 1 Algorithm 1 BiCGstab algorithm 1: Start with an initial guess x 2: Compute r = b Ax 3: Choose an arbitrary vector r such that ρ = (r r ), e.g., r = r 4: p = r 5: k= 6: repeat 7: q = Ap k 8: α = ρ k /(r q) 9: s = r k αq 1: t = As 11: ω = (t s)/(t t) 12: x k+1 = x k + αp k + ωs 13: r k+1 = s ωt 14: ρ k+1 = (r r k+1 ) 15: β = (α/ω)(ρ k+1 /ρ k ) 16: p k+1 = r k+1 + β(p k ωq) 17: ρ k ρ k+1 (a) Exchanging vectors. 18: k++ 19: until r k+1 2 / b 2 < ϵ (b) Reshaping arrays. Fig. 4: Two types of control methods. 2 Fig. 5: Rotating memory space of arrays. 5 Copyright c 215 by JSFM
6 4.1 SOR Bi-CGstab 1 Average time Fig. 6 Outer loop List 2 l loop if Inner loop(no) List 1 Inner loop(if) List 1 l loop if Inner loop(exchanging) Inner loop(reshaping) Outer loop Average time Inner loop Average time Inner loop Outer loop Average time [sec] Outer loop Inner loop(no) Inner loop(if) Inner loop(exchanging) Inner loop(reshaping) The number of RHS vectors (l) Fig. 6: Comparison of average time with a problem size Fig. 7 l Fig. 1(a) if 5. SPARC64 XIfx % JSPS (1),, (28) pp.1 7 (2) Samuel W. Williams, Andrew Waterman, David A. Patterson, Roofline: An Insightful Visual Performance Model for Floating Point Programs and Multicore Architectures, Commun, ACM, Vol. 52 (29), pp (3) Willcock, J. and Lumsdaine, A., Accelerating sparse matrix computations via data compression, Proc. 2th Annual ICS 6 (26) pp (4) Tang, W. T., et al., Accelerating Sparse Matrix-vector Multiplication on GPUs Using Bitrepresentation-optimized Schemes, Proc. of SC13 26 (213) pp.1 12 (5) Ono, K., Chiba, S., Inoue, S., Minami, K., Low Byte/Flop Implementation of Iterative Solver for Sparse Matrices Derived from Stencil Computations, High Performance Computing for Computational Science VECPAR 214 (215) Vol.8969 pp (6) Imamura, S., Ono, K., Yokokawa, M., PER- FORMANCE EVALUATION OF ITERATIVE SOLVER FOR MULTIPLE VECTORS ASSOCI- ATED WITH A LARGE-SCALE SPARSE MA- TRIX, Proc. 27 th International Conference on Parallel Computational Fluid Dynamics (215) pp (to appear) (7) Van der Vorst, H. A., Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems, SIAM J. Sci. and Stat. Comput. Vol.13 No.2 (1992) pp (8) Yokokawa, M., Vector-Parallel Processing of the Successive Overrelaxation Method, Japan Atomic Energy Research Institute JAERI-M Report No (1988) (in Japanese) (9) No If Exchanging Reshaping The number of RHS vectors (l) Fig. 7: Comparison of the performance of control methods with a problem size Copyright c 215 by JSFM
Performance improvement of iterative solver using bit-compression for a sparse matrix
E6- Performance improvement of iterative sover using -compression for a sparse matrix, 7--6, E-mai keno@riken.jp Kenji Ono, RIKE AICS, 7--6 Minatojima-minami-cho, Chuo-ku, Kobe, Japan A nobe Bit-representation/compression
Διαβάστε περισσότεραFX10 SIMD SIMD. [3] Dekker [4] IEEE754. a.lo. (SpMV Sparse matrix and vector product) IEEE754 IEEE754 [5] Double-Double Knuth FMA FMA FX10 FMA SIMD
FX,a),b),c) Bailey Double-Double [] FMA FMA [6] FX FMA SIMD Single Instruction Multiple Data 5 4.5. [] Bailey SIMD SIMD 8bit FMA (SpMV Sparse matrix and vector product) FX. DD Bailey Double-Double a) em49@ns.kogakuin.ac.jp
Διαβάστε περισσότεραBiCG CGS BiCGStab BiCG CGS 5),6) BiCGStab M Minimum esidual part CGS BiCGStab BiCGStab 2 PBiCG PCGS α β 3 BiCGStab PBiCGStab PBiCG 4 PBiCGStab 5 2. Bi
BiCGStab 1 1 2 3 1 4 2 BiCGStab PBiCGStab BiCG CGS CGS PBiCGStab BiCGStab M PBiCGStab An improvement in preconditioned algorithm of BiCGStab method Shoji Itoh, 1 aahiro Katagiri, 1 aao Saurai, 2 Mitsuyoshi
Διαβάστε περισσότεραGPU DD Double-Double 3 4 BLAS Basic Linear Algebra Subprograms [3] 2
GPU 4 1,a) 2,b) 1 GPU Tesla M2050 Double-Double DD 4 BiCGStab GPU 4 BiCGStab 1 1.0 2.2 4 GPU 4 1. IEEE754-2008[1] 128bit binary128 CG Conjugate Gradient [2] 1 1 2 a) mukunoki@hpcs.cs.tsukuba.ac.jp b) daisuke@cs.tsukuba.ac.jp
Διαβάστε περισσότεραES440/ES911: CFD. Chapter 5. Solution of Linear Equation Systems
ES440/ES911: CFD Chapter 5. Solution of Linear Equation Systems Dr Yongmann M. Chung http://www.eng.warwick.ac.uk/staff/ymc/es440.html Y.M.Chung@warwick.ac.uk School of Engineering & Centre for Scientific
Διαβάστε περισσότεραGPGPU. Grover. On Large Scale Simulation of Grover s Algorithm by Using GPGPU
GPGPU Grover 1, 2 1 3 4 Grover Grover OpenMP GPGPU Grover qubit OpenMP GPGPU, 1.47 qubit On Large Scale Simulation of Grover s Algorithm by Using GPGPU Hiroshi Shibata, 1, 2 Tomoya Suzuki, 1 Seiya Okubo
Διαβάστε περισσότεραGMRES(m) , GMRES, , GMRES(m), Look-Back GMRES(m). Ax = b, A C n n, x, b C n (1) Krylov.
211 9 12, GMRES,.,., Look-Back.,, Ax = b, A C n n, x, b C n (1),., Krylov., GMRES [5],.,., Look-Back [3]., 2 Krylov,. 3, Look-Back, 4. 5. 1 Algorith 1 The GMRES ethod 1: Choose the initial guess x and
Διαβάστε περισσότεραA Method for Creating Shortcut Links by Considering Popularity of Contents in Structured P2P Networks
P2P 1,a) 1 1 1 P2P P2P P2P P2P A Method for Creating Shortcut Links by Considering Popularity of Contents in Structured P2P Networks NARISHIGE Yuki 1,a) ABE Kota 1 ISHIBASHI Hayato 1 MATSUURA Toshio 1
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραA Fast Finite Element Electromagnetic Analysis on Multi-core Processer System
Vol. 3 No. 3 189 198 (Sep. 2010) 1 1 1 2 2 Arnold Folk Winther A Fast Finite Element Electromagnetic Analysis on Multi-core Processer System Takeshi Mifune, 1 Yu Hirotani, 1 Takeshi Iwashita, 1 Toshio
Διαβάστε περισσότεραGPU. CUDA GPU GeForce GTX 580 GPU 2.67GHz Intel Core 2 Duo CPU E7300 CUDA. Parallelizing the Number Partitioning Problem for GPUs
GPU 1 1 NP number partitioning problem Pedroso CUDA GPU GeForce GTX 580 GPU 2.67GHz Intel Core 2 Duo CPU E7300 CUDA C Pedroso Python 323 Python C 12.2 Parallelizing the Number Partitioning Problem for
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραBuried Markov Model Pairwise
Buried Markov Model 1 2 2 HMM Buried Markov Model J. Bilmes Buried Markov Model Pairwise 0.6 0.6 1.3 Structuring Model for Speech Recognition using Buried Markov Model Takayuki Yamamoto, 1 Tetsuya Takiguchi
Διαβάστε περισσότεραMatrices and vectors. Matrix and vector. a 11 a 12 a 1n a 21 a 22 a 2n A = b 1 b 2. b m. R m n, b = = ( a ij. a m1 a m2 a mn. def
Matrices and vectors Matrix and vector a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn def = ( a ij ) R m n, b = b 1 b 2 b m Rm Matrix and vectors in linear equations: example E 1 : x 1 + x 2 + 3x 4 =
Διαβάστε περισσότεραΑριθµητικές Μέθοδοι Collocation. Απεικόνιση σε Σύγχρονες Υπολογιστικές Αρχιτεκτονικές
Αριθµητικές Μέθοδοι Collocation Απεικόνιση σε Σύγχρονες Υπολογιστικές Αρχιτεκτονικές Hermite Collocation Method BVP L B uxy (, ) = f(, xy), (, xy) Ω uxy (, ) = gxy (, ), (, xy) Ω Red Black Collocation
Διαβάστε περισσότεραTMA4115 Matematikk 3
TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραBinary32 (a hi ) 8 bits 23 bits Binary32 (a lo ) 8 bits 23 bits Double-Float (a=a hi +a lo, a lo 0.5ulp(a hi ) ) 8 bits 46 bits Binary64 11 bits sign
Maxwell GPU DGEMM 1,a) 1,b) NVIDIA 2014 Maxwell GM107 GM204 GPU : =1:32 GM204 GeForce GTX 980 2 double-float DF BLAS DGEMM DGEMM DF DGEMM 2 1. IEEE 754-2008[1] binary32 binary64 NVIDIA GPU 2010 Fermi :
Διαβάστε περισσότεραWavelet based matrix compression for boundary integral equations on complex geometries
1 Wavelet based matrix compression for boundary integral equations on complex geometries Ulf Kähler Chemnitz University of Technology Workshop on Fast Boundary Element Methods in Industrial Applications
Διαβάστε περισσότεραRetrieval of Seismic Data Recorded on Open-reel-type Magnetic Tapes (MT) by Using Existing Devices
No. 3 + 1,**- Technical Research Report, Earthquake Research Institute, University of Tokyo, No. 3, pp. + 1,,**-. MT * ** *** Retrieval of Seismic Data Recorded on Open-reel-type Magnetic Tapes (MT) by
Διαβάστε περισσότεραRe-Pair n. Re-Pair. Re-Pair. Re-Pair. Re-Pair. (Re-Merge) Re-Merge. Sekine [4, 5, 8] (highly repetitive text) [2] Re-Pair. Blocked-Repair-VF [7]
Re-Pair 1 1 Re-Pair Re-Pair Re-Pair Re-Pair 1. Larsson Moffat [1] Re-Pair Re-Pair (Re-Pair) ( ) (highly repetitive text) [2] Re-Pair [7] Re-Pair Re-Pair n O(n) O(n) 1 Hokkaido University, Graduate School
Διαβάστε περισσότερα3: A convolution-pooling layer in PS-CNN 1: Partially Shared Deep Neural Network 2.2 Partially Shared Convolutional Neural Network 2: A hidden layer o
Sound Source Identification based on Deep Learning with Partially-Shared Architecture 1 2 1 1,3 Takayuki MORITO 1, Osamu SUGIYAMA 2, Ryosuke KOJIMA 1, Kazuhiro NAKADAI 1,3 1 2 ( ) 3 Tokyo Institute of
Διαβάστε περισσότεραEfficient Implementation of Sparse Linear Algebra Operations on InfiniBand Cluster. Akira Nishida,
InfiniBand,,.,, PCI Express InfiniBand,,.,,. Efficient Implementation of Sparse Linear Algebra Operations on InfiniBand Cluster Akira Nishida, Construction of scalable and low cost parallel computing environment
Διαβάστε περισσότεραDurbin-Levinson recursive method
Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again
Διαβάστε περισσότεραHigh order interpolation function for surface contact problem
3 016 5 Journal of East China Normal University Natural Science No 3 May 016 : 1000-564101603-0009-1 1 1 1 00444; E- 00030 : Lagrange Lobatto Matlab : ; Lagrange; : O41 : A DOI: 103969/jissn1000-56410160300
Διαβάστε περισσότεραDetection and Recognition of Traffic Signal Using Machine Learning
1 1 1 Detection and Recognition of Traffic Signal Using Machine Learning Akihiro Nakano, 1 Hiroshi Koyasu 1 and Hitoshi Maekawa 1 To improve road safety by assisting the driver, traffic signal recognition
Διαβάστε περισσότεραNew Adaptive Projection Technique for Krylov Subspace Method
1 2 New Adaptive Projection Technique for Krylov Subspace Method Akinori Kumagai 1 and Takashi Nodera 2 Generally projection technique in the numerical operation is one of the preconditioning commonly
Διαβάστε περισσότεραYoshifumi Moriyama 1,a) Ichiro Iimura 2,b) Tomotsugu Ohno 1,c) Shigeru Nakayama 3,d)
1,a) 2,b) 1,c) 3,d) Quantum-Inspired Evolutionary Algorithm 0-1 Search Performance Analysis According to Interpretation Methods for Dealing with Permutation on Integer-Type Gene-Coding Method based on
Διαβάστε περισσότερα(, ) (SEM) [4] ,,,, , Legendre. [6] Gauss-Lobatto-Legendre (GLL) Legendre. Dubiner ,,,, (TSEM) Vol. 34 No. 4 Dec. 2017
34 4 17 1 JOURNAL OF SHANGHAI POLYTECHNIC UNIVERSITY Vol. 34 No. 4 Dec. 17 : 11-4543(174-83-8 DOI: 1.1957/j.cnki.jsspu.17.4.6 (, 19 :,,,,,, : ; ; ; ; ; : O 41.8 : A, [1],,,,, Jung [] Legendre, [3] Chebyshev
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραQuick algorithm f or computing core attribute
24 5 Vol. 24 No. 5 Cont rol an d Decision 2009 5 May 2009 : 100120920 (2009) 0520738205 1a, 2, 1b (1. a., b., 239012 ; 2., 230039) :,,.,.,. : ; ; ; : TP181 : A Quick algorithm f or computing core attribute
Διαβάστε περισσότεραFourier transform, STFT 5. Continuous wavelet transform, CWT STFT STFT STFT STFT [1] CWT CWT CWT STFT [2 5] CWT STFT STFT CWT CWT. Griffin [8] CWT CWT
1,a) 1,2,b) Continuous wavelet transform, CWT CWT CWT CWT CWT 100 1. Continuous wavelet transform, CWT [1] CWT CWT CWT [2 5] CWT CWT CWT CWT CWT Irino [6] CWT CWT CWT CWT CWT 1, 7-3-1, 113-0033 2 NTT,
Διαβάστε περισσότεραIPSJ SIG Technical Report Vol.2014-CE-127 No /12/6 CS Activity 1,a) CS Computer Science Activity Activity Actvity Activity Dining Eight-He
CS Activity 1,a) 2 2 3 CS Computer Science Activity Activity Actvity Activity Dining Eight-Headed Dragon CS Unplugged Activity for Learning Scheduling Methods Hisao Fukuoka 1,a) Toru Watanabe 2 Makoto
Διαβάστε περισσότεραSimplex Crossover for Real-coded Genetic Algolithms
Technical Papers GA Simplex Crossover for Real-coded Genetic Algolithms 47 Takahide Higuchi Shigeyoshi Tsutsui Masayuki Yamamura Interdisciplinary Graduate school of Science and Engineering, Tokyo Institute
Διαβάστε περισσότερα: Monte Carlo EM 313, Louis (1982) EM, EM Newton-Raphson, /. EM, 2 Monte Carlo EM Newton-Raphson, Monte Carlo EM, Monte Carlo EM, /. 3, Monte Carlo EM
2008 6 Chinese Journal of Applied Probability and Statistics Vol.24 No.3 Jun. 2008 Monte Carlo EM 1,2 ( 1,, 200241; 2,, 310018) EM, E,,. Monte Carlo EM, EM E Monte Carlo,. EM, Monte Carlo EM,,,,. Newton-Raphson.
Διαβάστε περισσότεραGCG-type Methods for Nonsymmetric Linear Systems
GCG-type Methods for Nonsymmetric Linear Systems Tsung-Ming Huang Department of Mathematics National Taiwan Normal University December 6, 2011 T.M. Huang (NTNU) GCG-type Methods for Nonsymmetric LS December
Διαβάστε περισσότεραΜΕΘΟΔΟΙ ΑΕΡΟΔΥΝΑΜΙΚΗΣ
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ Εργαστήριο Θερμικών Στροβιλομηχανών Μονάδα Παράλληλης ης Υπολογιστικής Ρευστοδυναμικής & Βελτιστοποίησης ΜΕΘΟΔΟΙ ΑΕΡΟΔΥΝΑΜΙΚΗΣ ΒΕΛΤΙΣΤΟΠΟΙΗΣΗΣ (7 ο Εξάμηνο Σχολής Μηχ.Μηχ. ΕΜΠ)
Διαβάστε περισσότεραOrthogonalization Library with a Numerical Computation Policy Interface
Vol. 46 No. SIG 7(ACS 10) May 2005 DGKS PC 10 8 10 14 4.8 Orthogonalization Library with a Numerical Computation Policy Interface Ken Naono, Mitsuyoshi Igai and Hiroyuki Kidachi We propose an orthogonalization
Διαβάστε περισσότεραΕΚΤΙΜΗΣΗ ΤΟΥ ΚΟΣΤΟΥΣ ΤΩΝ ΟΔΙΚΩΝ ΑΤΥΧΗΜΑΤΩΝ ΚΑΙ ΔΙΕΡΕΥΝΗΣΗ ΤΩΝ ΠΑΡΑΓΟΝΤΩΝ ΕΠΙΡΡΟΗΣ ΤΟΥ
ΠΑΝΕΠΙΣΤΗΜΙΟ ΠΑΤΡΩΝ ΠΟΛΥΤΕΧΝΙΚΗ ΣΧΟΛΗ ΤΜΗΜΑ ΠΟΛΙΤΙΚΩΝ ΜΗΧΑΝΙΚΩΝ ΕΚΤΙΜΗΣΗ ΤΟΥ ΚΟΣΤΟΥΣ ΤΩΝ ΟΔΙΚΩΝ ΑΤΥΧΗΜΑΤΩΝ ΚΑΙ ΔΙΕΡΕΥΝΗΣΗ ΤΩΝ ΠΑΡΑΓΟΝΤΩΝ ΕΠΙΡΡΟΗΣ ΤΟΥ ΔΙΑΤΡΙΒΗ ΔΙΠΛΩΜΑΤΟΣ ΕΙΔΙΚΕΥΣΗΣ ΔΗΜΗΤΡΙΟΥ Ν. ΠΙΤΕΡΟΥ
Διαβάστε περισσότεραA Hierarchy of Theta Bodies for Polynomial Systems
A Hierarchy of Theta Bodies for Polynomial Systems Rekha Thomas, U Washington, Seattle Joint work with João Gouveia (U Washington) Monique Laurent (CWI) Pablo Parrilo (MIT) The Theta Body of a Graph G
Διαβάστε περισσότεραComputing Gradient. Hung-yi Lee 李宏毅
Computing Gradient Hung-yi Lee 李宏毅 Introduction Backpropagation: an efficient way to compute the gradient Prerequisite Backpropagation for feedforward net: http://speech.ee.ntu.edu.tw/~tkagk/courses/mlds_05_/lecture/
Διαβάστε περισσότερα[4] 1.2 [5] Bayesian Approach min-max min-max [6] UCB(Upper Confidence Bound ) UCT [7] [1] ( ) Amazons[8] Lines of Action(LOA)[4] Winands [4] 1
1,a) Bayesian Approach An Application of Monte-Carlo Tree Search Algorithm for Shogi Player Based on Bayesian Approach Daisaku Yokoyama 1,a) Abstract: Monte-Carlo Tree Search (MCTS) algorithm is quite
Διαβάστε περισσότεραAn Automatic Modulation Classifier using a Frequency Discriminator for Intelligent Software Defined Radio
C IEEJ Transactions on Electronics, Information and Systems Vol.133 No.5 pp.910 915 DOI: 10.1541/ieejeiss.133.910 a) An Automatic Modulation Classifier using a Frequency Discriminator for Intelligent Software
Διαβάστε περισσότεραΤΕΧΝΙΚΕΣ ΑΥΞΗΣΗΣ ΤΗΣ ΑΠΟΔΟΣΗΣ ΤΩΝ ΥΠΟΛΟΓΙΣΤΩΝ I
ΤΕΧΝΙΚΕΣ ΑΥΞΗΣΗΣ ΤΗΣ ΑΠΟΔΟΣΗΣ ΤΩΝ ΥΠΟΛΟΓΙΣΤΩΝ I MIPS Η MIPS (Microprocessor without Interlocked Pipeline Stages) είναι μία αρχιτεκτονική συνόλου εντολών (ISA) γλώσσας μηχανής που αναπτύχθηκε από την εταιρεία
Διαβάστε περισσότεραComputation Method to Improve Three-phase Voltage Imbalance by Exchange of Single-phase Load Connection
Computation Method to Improve Three-phase Voltage Imbalance by Exchange of Single-phase Load Connection Yasuhiro Hayashi, Member, Junya Matsuki, Member, Masayoshi Ohashi, Student Member, Yasuyuki Tada,Member
Διαβάστε περισσότεραΕΘΝΙΚΟ ΚΑΙ ΚΑΠΟΔΙΣΤΡΙΑΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΘΗΝΩΝ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΚΑΙ ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ
ΕΘΝΙΚΟ ΚΑΙ ΚΑΠΟΔΙΣΤΡΙΑΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΘΗΝΩΝ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΚΑΙ ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ "ΤΕΧΝΟΛΟΓΙΕΣ ΠΛΗΡΟΦΟΡΙΚΗΣ ΣΤΗΝ ΙΑΤΡΙΚΗ ΚΑΙ ΤΗ ΒΙΟΛΟΓΙΑ" ΔΙΠΛΩΜΑΤΙΚΗ
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραHOSVD. Higher Order Data Classification Method with Autocorrelation Matrix Correcting on HOSVD. Junichi MORIGAKI and Kaoru KATAYAMA
DEIM Forum 2010 D1-4 HOSVD 191-0065 6-6 E-mail: j.morigaki@gmail.com, katayama@tmu.ac.jp Lathauwer (HOSVD) (Tensor) HOSVD Savas HOSVD Sun HOSVD,, Higher Order Data Classification Method with Autocorrelation
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραMaude 6. Maude [1] UIUC J. Meseguer. Maude. Maude SRI SRI. Maude. AC (Associative-Commutative) Maude. Maude Meseguer OBJ LTL SPIN
78 Maude 1 Maude [1] UIUC J. Meseguer ( 1 ) ( ) Maude Maude SRI 90 UIUC SRI Maude SRI S. Eker C++ Maude 2 Maude Meseguer OBJ 1983-84 OBJ2[3] OBJ Maude OBJ 1 CafeOBJ 3 Maude 4 Maude CafeOBJ Maude: A Computer
Διαβάστε περισσότεραkatoh@kuraka.co.jp okaken@kuraka.co.jp mineot@fukuoka-u.ac.jp 4 35 3 Normalized stress σ/g 25 2 15 1 5 Breaking test Theory 1 2 Shear tests Failure tests Compressive tests 1 2 3 4 5 6 Fig.1. Relation between
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραER-Tree (Extended R*-Tree)
1-9825/22/13(4)768-6 22 Journal of Software Vol13, No4 1, 1, 2, 1 1, 1 (, 2327) 2 (, 3127) E-mail xhzhou@ustceducn,,,,,,, 1, TP311 A,,,, Elias s Rivest,Cleary Arya Mount [1] O(2 d ) Arya Mount [1] Friedman,Bentley
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραComparison of Evapotranspiration between Indigenous Vegetation and Invading Vegetation in a Bog
J. Jpn. Soc. Soil Phys. No. +*-, p.-3.1,**0 ** * *** Comparison of Evapotranspiration between Indigenous Vegetation and Invading Vegetation in a Bog Toshiki FUJIMOTO*, Ippei IIYAMA*, Mai SAKAI*, Osamu
Διαβάστε περισσότεραMetal Oxide Varistors (MOV) Data Sheet
Φ SERIES Metal Oxide Varistors (MOV) Data Sheet Features Wide operating voltage (V ma ) range from 8V to 0V Fast responding to transient over-voltage Large absorbing transient energy capability Low clamping
Διαβάστε περισσότεραBundle Adjustment for 3-D Reconstruction: Implementation and Evaluation
3 2 3 2 3 undle Adjustment or 3-D Reconstruction: Implementation and Evaluation Yuuki Iwamoto, Yasuyuki Sugaya 2 and Kenichi Kanatani We describe in detail the algorithm o bundle adjustment or 3-D reconstruction
Διαβάστε περισσότεραΠρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής
Πρόβλημα 1: Αναζήτηση Ελάχιστης/Μέγιστης Τιμής Να γραφεί πρόγραμμα το οποίο δέχεται ως είσοδο μια ακολουθία S από n (n 40) ακέραιους αριθμούς και επιστρέφει ως έξοδο δύο ακολουθίες από θετικούς ακέραιους
Διαβάστε περισσότεραECE 468: Digital Image Processing. Lecture 8
ECE 468: Digital Image Processing Lecture 8 Prof. Sinisa Todorovic sinisa@eecs.oregonstate.edu 1 Image Reconstruction from Projections X-ray computed tomography: X-raying an object from different directions
Διαβάστε περισσότερα1 (forward modeling) 2 (data-driven modeling) e- Quest EnergyPlus DeST 1.1. {X t } ARMA. S.Sp. Pappas [4]
212 2 ( 4 252 ) No.2 in 212 (Total No.252 Vol.4) doi 1.3969/j.issn.1673-7237.212.2.16 STANDARD & TESTING 1 2 2 (1. 2184 2. 2184) CensusX12 ARMA ARMA TU111.19 A 1673-7237(212)2-55-5 Time Series Analysis
Διαβάστε περισσότεραHIV HIV HIV HIV AIDS 3 :.1 /-,**1 +332
,**1 The Japanese Society for AIDS Research The Journal of AIDS Research +,, +,, +,, + -. / 0 1 +, -. / 0 1 : :,**- +,**. 1..+ - : +** 22 HIV AIDS HIV HIV AIDS : HIV AIDS HIV :HIV AIDS 3 :.1 /-,**1 HIV
Διαβάστε περισσότερα- 1+x 2 - x 3 + 7x4. 40 + 127x8. 12 - x5 4 + 31x6. 360 - x 7. - 1+x 2 - x 3 - -
a.bergara@ehu.es - 1 x 2 - - - - - - - Ο - 1x 2 - x 3 - - - - - - 1 x 2 - x 3 7 x4 12-1x 2 - x 3 7x4 12 - x5 4 31x6 360 - x 7 40 127x8 20160 - - - Ο clear; % Coefficients of the equation: a x'b x c
Διαβάστε περισσότεραΓΕΩΜΕΣΡΙΚΗ ΣΕΚΜΗΡΙΩΗ ΣΟΤ ΙΕΡΟΤ ΝΑΟΤ ΣΟΤ ΣΙΜΙΟΤ ΣΑΤΡΟΤ ΣΟ ΠΕΛΕΝΔΡΙ ΣΗ ΚΤΠΡΟΤ ΜΕ ΕΦΑΡΜΟΓΗ ΑΤΣΟΜΑΣΟΠΟΙΗΜΕΝΟΤ ΤΣΗΜΑΣΟ ΨΗΦΙΑΚΗ ΦΩΣΟΓΡΑΜΜΕΣΡΙΑ
ΕΘΝΙΚΟ ΜΕΣΟΒΙΟ ΠΟΛΤΣΕΧΝΕΙΟ ΣΜΗΜΑ ΑΓΡΟΝΟΜΩΝ-ΣΟΠΟΓΡΑΦΩΝ ΜΗΧΑΝΙΚΩΝ ΣΟΜΕΑ ΣΟΠΟΓΡΑΦΙΑ ΕΡΓΑΣΗΡΙΟ ΦΩΣΟΓΡΑΜΜΕΣΡΙΑ ΓΕΩΜΕΣΡΙΚΗ ΣΕΚΜΗΡΙΩΗ ΣΟΤ ΙΕΡΟΤ ΝΑΟΤ ΣΟΤ ΣΙΜΙΟΤ ΣΑΤΡΟΤ ΣΟ ΠΕΛΕΝΔΡΙ ΣΗ ΚΤΠΡΟΤ ΜΕ ΕΦΑΡΜΟΓΗ ΑΤΣΟΜΑΣΟΠΟΙΗΜΕΝΟΤ
Διαβάστε περισσότεραLifting Entry (continued)
ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu
Διαβάστε περισσότεραn 1 n 3 choice node (shelf) choice node (rough group) choice node (representative candidate)
THE INSTITUTE OF ELECTRONICS, INFORMATION AND COMMUNICATION ENGINEERS TECHNICAL REPORT OF IEICE. y y yy y 1565 0871 2 1 yy 525 8577 1 1 1 E-mail: yfmakihara,shiraig@cv.mech.eng.osaka-u.ac.jp, yyshimada@ci.ritsumei.ac.jp
Διαβάστε περισσότεραΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΠΟΛΙΤΙΚΩΝ ΜΗΧΑΝΙΚΩΝ. «Θεσμικό Πλαίσιο Φωτοβολταïκών Συστημάτων- Βέλτιστη Απόδοση Μέσω Τρόπων Στήριξης»
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΠΟΛΙΤΙΚΩΝ ΜΗΧΑΝΙΚΩΝ ΤΟΜΕΑΣ ΑΝΘΡΩΠΙΣΤΙΚΩΝ & ΚΟΙΝΩΝΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΚΑΙ ΔΙΚΑΙΟΥ «Θεσμικό Πλαίσιο Φωτοβολταïκών Συστημάτων- Βέλτιστη Απόδοση Μέσω Τρόπων Στήριξης» Διπλωματική
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραStudy of In-vehicle Sound Field Creation by Simultaneous Equation Method
Study of In-vehicle Sound Field Creation by Simultaneous Equation Method Kensaku FUJII Isao WAKABAYASI Tadashi UJINO Shigeki KATO Abstract FUJITSU TEN Limited has developed "TOYOTA remium Sound System"
Διαβάστε περισσότεραSupplementary Materials for Evolutionary Multiobjective Optimization Based Multimodal Optimization: Fitness Landscape Approximation and Peak Detection
IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. XX, NO. X, XXXX XXXX Supplementary Materials for Evolutionary Multiobjective Optimization Based Multimodal Optimization: Fitness Landscape Approximation
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραBreaking capacity: ~200kA Rated voltage: ~690V, 550V. Operating I 2 t-value (A 2 s) Power
SYSTEM NV-NH NV/NH SERIES TYPES gr UQ M M, M-striker pin ~ 5V ~9V Technical data on page 8 Technical data: Application: MCUQ/5A/9V Standards: IEC 9- Breaking capacity: ~ka Rated voltage: ~9V, 55V For battery
Διαβάστε περισσότεραSMD Transient Voltage Suppressors
SMD Transient Suppressors Feature Full range from 0 to 22 series. form 4 to 60V RMS ; 5.5 to 85Vdc High surge current ability Bidirectional clamping, high energy Fast response time
Διαβάστε περισσότεραOptimization, PSO) DE [1, 2, 3, 4] PSO [5, 6, 7, 8, 9, 10, 11] (P)
( ) 1 ( ) : : (Differential Evolution, DE) (Particle Swarm Optimization, PSO) DE [1, 2, 3, 4] PSO [5, 6, 7, 8, 9, 10, 11] 2 2.1 (P) (P ) minimize f(x) subject to g j (x) 0, j = 1,..., q h j (x) = 0, j
Διαβάστε περισσότεραLaplace Expansion. Peter McCullagh. WHOA-PSI, St Louis August, Department of Statistics University of Chicago
Laplace Expansion Peter McCullagh Department of Statistics University of Chicago WHOA-PSI, St Louis August, 2017 Outline Laplace approximation in 1D Laplace expansion in 1D Laplace expansion in R p Formal
Διαβάστε περισσότεραJ. of Math. (PRC) 6 n (nt ) + n V = 0, (1.1) n t + div. div(n T ) = n τ (T L(x) T ), (1.2) n)xx (nt ) x + nv x = J 0, (1.4) n. 6 n
Vol. 35 ( 215 ) No. 5 J. of Math. (PRC) a, b, a ( a. ; b., 4515) :., [3]. : ; ; MR(21) : 35Q4 : O175. : A : 255-7797(215)5-15-7 1 [1] : [ ( ) ] ε 2 n n t + div 6 n (nt ) + n V =, (1.1) n div(n T ) = n
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραUsing the Jacobian- free Newton- Krylov method to solve the sea- ice momentum equa<on
Using the Jacobian- free Newton- Krylov method to solve the sea- ice momentum equa
Διαβάστε περισσότεραΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΟΔΟΝΤΙΚΗΣ ΚΑΙ ΑΝΩΤΕΡΑΣ ΠΡΟΣΘΕΤΙΚΗΣ
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΟΔΟΝΤΙΑΤΡΙΚΗΣ ΕΡΓΑΣΤΗΡΙΟ ΟΔΟΝΤΙΚΗΣ ΚΑΙ ΑΝΩΤΕΡΑΣ ΠΡΟΣΘΕΤΙΚΗΣ ΣΥΓΚΡΙΤΙΚΗ ΜΕΛΕΤΗ ΤΗΣ ΣΥΓΚΡΑΤΗΤΙΚΗΣ ΙΚΑΝΟΤΗΤΑΣ ΟΡΙΣΜΕΝΩΝ ΠΡΟΚΑΤΑΣΚΕΥΑΣΜΕΝΩΝ ΣΥΝΔΕΣΜΩΝ ΑΚΡΙΒΕΙΑΣ
Διαβάστε περισσότεραΜΕΛΕΤΗ ΑΠΟΤΙΜΗΣΗΣ ΠΑΡΑΓΟΜΕΝΗΣ ΕΝΕΡΓΕΙΑΣ Φ/Β ΣΥΣΤΗΜΑΤΩΝ ΕΓΚΑΤΕΣΤΗΜΕΝΩΝ ΣΕ ΕΛΛΑ Α ΚΑΙ ΤΣΕΧΙΑ
ΜΕΛΕΤΗ ΑΠΟΤΙΜΗΣΗΣ ΠΑΡΑΓΟΜΕΝΗΣ ΕΝΕΡΓΕΙΑΣ Φ/Β ΣΥΣΤΗΜΑΤΩΝ ΕΓΚΑΤΕΣΤΗΜΕΝΩΝ ΣΕ ΕΛΛΑ Α ΚΑΙ ΤΣΕΧΙΑ V. Poulek 1, M. Libra 2, Ι. Γράβαλος 3, Θ. Γιαλαµάς 3,. Μόσχου 4, Π. Ξυραδάκης 3,. Κατέρης 4, Ζ. Τσιρόπουλος 3,
Διαβάστε περισσότεραΕφαρµογές µε ανάγκες για υψηλές επιδόσεις Αξιολόγηση επίδοσης Παραλληλοποίηση εφαρµογών
Parallelizing applications for the GRID Γιώργος Γκούµας goumas@cslab.ece.ntua.gr Σύνοψη Παρουσίασηςασης Εφαρµογές µε ανάγκες για υψηλές επιδόσεις Αξιολόγηση επίδοσης Παραλληλοποίηση εφαρµογών Γενικές αρχές
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 24/3/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Όλοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα μικρότεροι του 10000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Αν κάπου κάνετε κάποιες υποθέσεις
Διαβάστε περισσότεραCUDA FFT. High Performance 3-D FFT in CUDA Environment. Akira Nukada, 1, 2 Yasuhiko Ogata, 1, 2 Toshio Endo 1, 2 and Satoshi Matsuoka 1, 2, 3
Vol. 1 No. 2 231 239 (Aug. 2008) CUDA 3 FFT 1, 2 1, 2 1, 2 1, 2, 3 NVIDIA GPU CUDA CUDA 3 FFT GeForce 8 GPU 3 FFT CUFFT 1.1 3.1 3.3 79.5 GFLOPS High Performance 3-D FFT in CUDA Environment Akira Nukada,
Διαβάστε περισσότεραA research on the influence of dummy activity on float in an AOA network and its amendments
2008 6 6 :100026788 (2008) 0620106209,, (, 102206) : NP2hard,,..,.,,.,.,. :,,,, : TB11411 : A A research on the influence of dummy activity on float in an AOA network and its amendments WANG Qiang, LI
Διαβάστε περισσότεραDEIM Forum 2016 G7-5 152-8565 2-12-1 152-8565 2-12-1 889-1601 5200 E-mail: uragaki.k.aa@m.titech.ac.jp,,,.,,,,,,, 1. 1. 1,,,,,,.,,,,, 1. 2 [1],,,,, [2] (, SPM),,,,,,,. [3],, [4]. 2 A,B, A B, B A, B, 2,,,
Διαβάστε περισσότεραMetal Film Flame-Proof Resistors
Features -Low Noise -Low TCR from ±15~100PPM/ -High Precision from ±0.1%~1% -Complete Flameproof Construction UL-1412. -Coating meets UL94V-0 Applications -Telecommunication -Medical Equipment -Consumer
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραCorrection of chromatic aberration for human eyes with diffractive-refractive hybrid elements
5 5 2012 10 Chinese Optics Vol. 5 No. 5 Oct. 2012 1674-2915 2012 05-0525-06 - * 100190-14 - - 14. 51 μm 81. 4 μm - 1. 64 μm / O436. 1 TH703 A doi 10. 3788 /CO. 20120505. 0525 Correction of chromatic aberration
Διαβάστε περισσότεραA Method of Trajectory Tracking Control for Nonminimum Phase Continuous Time Systems
IIC-11-8 A Method of Trajectory Tracking Control for Nonminimum Phase Continuous Time Systems Takayuki Shiraishi, iroshi Fujimoto (The University of Tokyo) Abstract The purpose of this paper is achievement
Διαβάστε περισσότεραdifluoroboranyls derived from amides carrying donor group Supporting Information
The influence of the π-conjugated spacer on photophysical properties of difluoroboranyls derived from amides carrying donor group Supporting Information Anna Maria Grabarz a Adèle D. Laurent b, Beata Jędrzejewska
Διαβάστε περισσότεραConjoint. The Problems of Price Attribute by Conjoint Analysis. Akihiko SHIMAZAKI * Nobuyuki OTAKE
Conjoint Conjoint The Problems of Price Attribute by Conjoint Analysis Akihiko SHIMAZAKI * Nobuyuki OTAKE +, Conjoint - Conjoint. / 0 PSM Price Sensitivity Measurement Conjoint 1 2 + Conjoint Luce and
Διαβάστε περισσότεραDevelopment of a Tiltmeter with a XY Magnetic Detector (Part +)
No. 2 +0,/,**, Technical Research Report, Earthquake Research Institute, University of Tokyo, No. 2, pp.+0,/,,**,. XY * * ** *** **** ***** Development of a Tiltmeter with a XY Magnetic Detector (Part
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραCORDIC Background (4A)
CORDIC Background (4A Copyright (c 20-202 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότερα