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Loan C. Ч Computational Frameworks for the Fast Fourier Transform (Frontiers in Applied Mathematics)
Loan C. Ч Computational Frameworks for the Fast Fourier Transform (Frontiers in Applied Mathematics)

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Ќазвание: Computational Frameworks for the Fast Fourier Transform (Frontiers in Applied Mathematics)

јвтор: Loan C.

јннотаци€:

The most comprehensive treatment of FFTs to date. Van Loan captures the interplay between mathematics and the design of effective numerical algorithms Ч a critical connection as more advanced machines become available. A stylized Matlab notation, which is familiar to those engaged in high-performance computing, is used. The Fast Fourier Transform (FFT) family of algorithms has revolutionized many areas of scientific computation. The FFT is one of the most widely used algorithms in science and engineering, with applications in almost every discipline. This volume is essential for professionals interested in linear algebra as well as those working with numerical methods. The FFT is also a great vehicle for teaching key aspects of scientific computing.


язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1992

 оличество страниц: 287

ƒобавлена в каталог: 21.02.2015

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
Array interpretation, bit reversal      42Ч43
Array interpretation, Cooley Ч Tukey      46Ч48
Array interpretation, index reversal      88Ч90
Array interpretation, radix-p butterfly      82
Array interpretation, transposed Stockham framework      53
Array/vector duality      5Ч6
Barrier construct      178Ч179
Bit reversal of index      39
Bit reversal via even-odd sort      42
Bit reversal via perfect shuffles      41
Bit reversal, distributed-memory      162Ч165
Bit reversal, permutation      39
Block matrices      5
Blocking permutation      192
Bluestein chirp-z algorithm      210
Boundary conditions, Dirichlet      249Ч250
Boundary conditions, Dirichlet Ч Neumann      250Ч251
Boundary conditions, Neumann-Neumann      252
Boundary conditions, periodic      253
Butterfly, conjugate even      218ff
Butterfly, Cooley Ч Tukey, definition of      70
Butterfly, Cooley Ч Tukey, radix-2 setting      28Ч30
Butterfly, Cooley Ч Tukey, s-processor      160Ч162
Butterfly, Cooley Ч Tukey, two-processor      158
Butterfly, Gentleman Ч Sande, definition of      71
Butterfly, Gentleman Ч Sande, two-processor      169
Butterfly, Hartley      225
Butterfly, matrix      18
Butterfly, radix-2      18
Butterfly, radix-8      107Ч109
Butterfly, radix-p      79
Butterfly, split-radix      113
Butterfly, vectorization of      34
Cache, definition      127
Cache, line      128
Cache, miss      129
Chinese remainder theorem mapping      192
Circulant matrices, properties      206Ч207
Colon notation      4
Column partitioning      4
Communication overhead      159Ч160
Complex arithmetic, flops and      9
Complex arithmetic, three-multiply version      10
Computation tree, mixed-radix      82
Computation tree, radix-2      13Ч14
Computation tree, split-radix      114
Computation tree, symmetric FFT      244
Conjugate even, butterfly      218
Conjugate even, DFTs      222Ч223
Conjugate even, vectors, data structure      216ff 234
Conjugate odd vectors      234
Convolution      205ff
Convolution, FFTs and      207Ч208
Convolution, two-dimensional      212Ч214
Cooley Ч Tukey, combine phase      21
Cooley Ч Tukey, distributed-memory      165Ч168
Cooley Ч Tukey, mixed-radix      95Ч98
Cooley Ч Tukey, multiple DFTs and      123
Cooley Ч Tukey, permutation phase      21
Cooley Ч Tukey, radix-2, discussion of      44ff
Cooley Ч Tukey, radix-2, flop analysis      45Ч46
Correlation      214
Cosine matrix, definition      230
Cosine matrix, scaled      232
Cosine transform (discrete), definition      229
Cosine transform (discrete), inverse of      240
Cosine transform (discrete), matrix specification      230 231 238Ч239
Cosine transform-II (discrete), definition      229
Cosine transform-II (discrete), inverse of      243
Cosine transform-II (discrete), matrix specification      230 233 242Ч243
Data motion overheads, shared-memory      177
Data re-use, radix and      109Ч110
Decimation in frequency      68
Decimation in frequency for real data      223
Decimation in time (DIT)      67
Diagonal scaling      7
Dirichlet boundary conditions      249Ч250
Dirichlet Ч Neumann boundary conditions      250Ч251
Discrete cosine transform (DCT)      238ff
Discrete cosine transform-II (DCT-II)      242ff
Discrete Fourier Transform (DFT) of structured vectors      234
Discrete Fourier Transform (DFT), definition      2
Discrete Fourier transform (DFT), matrix      3
Discrete sine transform (DST)      236ff
Discrete sine transform-II (DST-II)      240ff
Downshift permutation      206
Dynamic scheduling      186
Edson factorization      220
Edson real-data FFT      220ff
Even-odd sort permutation      12
Exchange matrix      215
Factorizations, Bluestein chirp-z      209Ч210
Factorizations, Cooley Ч Tukey (mixed-radix)      96Ч98
Factorizations, Cooley Ч Tukey (radix-2)      17ff
Factorizations, decimation in frequency (DIF)      64Ч65
Factorizations, Edson      220
Factorizations, Hartley      225
Factorizations, Pease (mixed-radix)      96 99Ч100
Factorizations, Pease (radix-2)      62Ч63
Factorizations, prime factor      196
Factorizations, Rader      211
Factorizations, rotated DFT      198Ч201
Factorizations, split radix      116
Factorizations, Stockham (mixed-radix)      96 100Ч101
Factorizations, Stockham (radix-2)      55
Factorizations, transposed Stockham (mixed-radix)      96 98
Factorizations, transposed Stockham (radix-2)      50Ч51
Factorizations, Winograd      201Ч202
Fast Fourier Transform Frameworks, (DIF) Cooley Ч Tukey      67
Fast Fourier Transform Frameworks, (DIF) Pease      68
Fast Fourier Transform Frameworks, (DIF) Stockham      69
Fast Fourier Transform Frameworks, (DIF) transposed Stockham      69
Fast Fourier Transform Frameworks, blocked four-step      144
Fast Fourier Transform Frameworks, blocked six-step      143
Fast Fourier Transform Frameworks, Bluestein chirp-z      210
Fast Fourier Transform Frameworks, Cooley Ч Tukey radix-2, in-place      44
Fast Fourier Transform Frameworks, Cooley Ч Tukey radix-2, unit stride      45
Fast Fourier Transform Frameworks, Cooley Ч Tukey radix-4      104
Fast Fourier Transform Frameworks, d-dimensional      152Ч154
Fast Fourier Transform Frameworks, distributed-memory      167
Fast Fourier Transform Frameworks, external memory      133Ч134
Fast Fourier Transform Frameworks, four-step      140
Fast Fourier Transform Frameworks, general radix      81
Fast Fourier Transform Frameworks, Pease radix-2      62
Fast Fourier Transform Frameworks, prime factor      195Ч196
Fast Fourier Transform Frameworks, Rader      212
Fast Fourier Transform Frameworks, rotated prime-factor      201
Fast Fourier Transform Frameworks, shared memory      180Ч181
Fast Fourier Transform Frameworks, six-step      140
Fast Fourier Transform Frameworks, split-radix      115 118
Fast Fourier Transform Frameworks, Stockham radix-2      56Ч57
Fast Fourier Transform Frameworks, Stockham radix-4      106
Fast Fourier Transform Frameworks, Stockham vector radix      149
Fast Fourier Transform Frameworks, three-dimensional transform      151
Fast Fourier Transform Frameworks, transposed Stockham radix-2      52
Fast Fourier Transform Frameworks, Winograd      201Ч202
Fast Fourier Transform, main idea of      11ff
Fast Poisson solvers      256Ч257
FLOP      9
Four-step framework, blocked      144
Four-step framework, distributed memory      173
Four-step framework, read data      224
Four-step framework, shared memory      184
Fraser transpose      133Ч138
General radix framework, ideas behind      76ff
General radix framework, recursive specification      81
Gentleman Ч Sande butterfly      66
Gentleman Ч Sande framework      67
Gentleman Ч Sande idea      65Ч66
Givens rotations      24Ч25
Hankel matrix      214
Hartley butterfly      225
Hartley factorization      225
Hartley transform      224ff
Hartley transform framework      227
History of the FFT      xiЧxii 16
In-place approach      44
Index-reversal, algorithm for      91
Index-reversal, array interpretation      88Ч90
Index-reversal, example      91
Index-reversal, permutation      87
Index-reversal, symmetry and      92
Intermediate DFTs, notion of      15
Intermediate DFTs, Stockham setting      49
Intermediate DFTs, storage schemes      95
Inverse FFT framework      64 69ff
Inverse real periodic transform, computation of      235Ч236
Inverse real periodic transform, definition      229
Kron1      7
Kron11      85
Kron12      86
Kron2      7
Kron3      8
Kron4      8
Kron5      8
Kron6      8
Kron7      8
Kron8      84
Kron9      84
Kronecker product definition      7
Kronecker product definition, properties      84Ч86
Long weight vector      34Ч35
Loop re-ordering      28Ч30 122Ч125
Matlab notation      9
Matrices, bit-reversal      20Ч21
Matrices, block      5
Matrices, butterfly      18
Matrices, complex-symmetric      3
Matrices, conjugate even butterfly      218
Matrices, cosine      230
Matrices, diagonal      7
Matrices, discrete Fourier transform      3
Matrices, even-odd sort permutation      12 39ff
Matrices, Hartley butterfly      225
Matrices, index-reversal permutation      87
Matrices, intermediate DFT      15
Matrices, mod p sort permutation      77
Matrices, orthogonal      3
Matrices, perfect shuffle      40ff
Matrices, permutation      6
Matrices, radix-p butterfly      79
Matrices, sine      230
Matrices, split radix butterfly      113
Matrices, symmetric      3
Matrices, unitary      3
Matrix, conjugate      2
Matrix, conjugate transpose      3
Matrix, transpose      2
Memory bank conflict in transposition      126
Memory bank conflict, illustration of      32
Memory references      42
Message identifier      157
Mixed-radix factorizations, Cooley Ч Tukey      97Ч98
Mixed-radix factorizations, Pease      99Ч100
Mixed-radix factorizations, Stockham      100Ч101
Mixed-radix factorizations, summary      96
Mixed-radix factorizations, transposed Stockham      98
Mixed-radix framework      82
Mixed-radix ideas      95ff
mod p, perfect shuffle      78
mod p, sort      77Ч78
Multidimensional, DFT      148
Multidimensional, FFT      152Ч154
Multiple transforms, approaches to      122ff
Multiple transforms, blocking      141
Multiple transforms, general radix setting and      81
Multiple transforms, multicolumn, definition of      122
Multiple transforms, multirow with small buffer      145
Multiple transforms, multirow, definition of      122
Multiple transforms, shared-memory      177Ч178
Neumann Ч Neumann boundary conditions      252Ч253
Node program      156
Notation, colon      4
Notation, submatrix designation      4
Pease FFT, mixed-radix      95Ч96 99Ч100
Pease FFT, radix-2      60ff
Perfect shuffle, definition      40Ч41
Perfect shuffle, mod p      78
Perfect shuffle, Stockham context      51Ч52
Periodic boundary conditions      253
Permutation by cycles      92Ч93
Permutations, bit reversal      36Ч39
Permutations, definition of      6
Permutations, downshift      206
Permutations, exchange      215
Permutations, index-reversal      87
Permutations, reflection matrix      215
Pipelining      32
Pointwise multiplication      7
Poisson equation on rectangle      254
Poisson equation, one-dimensional      247
Pool-of-task paradigm      184Ч186
Prime factor, algorithm      195Ч196
Prime factor, splitting      195
Primitive root      210
Quarter-wave even vectors      234
Quarter-wave odd vectors      234
Rader FFT      212
Radix, re-use of data and      109Ч110
Radix-2, computation tree      13Ч14
Radix-2, recursive specification      16
Radix-4, butterfly      102Ч104
Radix-4, Cooley Ч Tukey framework      104
Radix-4, Stockham framework      106
Radix-p      82
Radix-p splitting      78Ч79
Read/write overheads      177
Real arithmetic      32Ч33
Real data FFTs, autosort frameworks      221Ч222
Real data FFTs, discussion of      215ff
Real even vectors      234
Real odd vectors      234
Recursive algorithms, general radix      81
Recursive algorithms, radix-2      16
Recursive algorithms, split-radix      115
Recursive algorithms, symmetric FFT      245
recv      157
Reflection matrix      215
Relative primality      189Ч191
Ring      156
Rotated DFTs      197Ч201
Row partitioning      4
Ruritanian mapping      194Ч195
Scheduling dynamic      186
send      157
Shared-memory system      176
Signal flow graph      70
Sine matrix, definition      230
Sine matrix, scaled      232
Sine transform (discrete), definition      229
Sine transform (discrete), inverse of      240
Sine transform (discrete), matrix specification      230 231 236Ч238
Sine transform-II (discrete), definition      229
Sine transform-II (discrete), inverse of      241Ч242
Sine transform-II (discrete), matrix specification      230 233 240Ч241
Sine/cosine blocking      231
Six-point DFT      202Ч204
Six-step framework, blocked      143
Six-step framework, definition of      140
Six-step framework, distributed-memory      173
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