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Cholesky Factorization
5-78
5Cholesky Factorization
Purpose Factor a square Hermitian positive definite matrix into triangular
components.
Library Math Functions / Matrices and Linear Algebra / Matrix Factorizations
Description The Cholesky Factorization block uniquely factors the square Hermitian
positive definite input matrix S as
where L is a lower triangular square matrix with positive diagonal elements
and L
*
is the Hermitian (complex conjugate) transpose of L. Only the diagonal
and upper triangle of the input matrix are used, and any imaginary component
of the diagonal entries is disregarded.
The block’s output is a composite matrix with lower triangle elements from L
and upper triangle elements from L
*
, and is always sample-based.
Note that L and L
*
share the same diagonal in the output matrix. Cholesky
factorization requires half the computation of Gaussian elimination
(LU decomposition), and is always stable.
The algorithm requires that the input be square and Hermitian positive
definite. When the input is not positive definite, the block reacts with the
behavior specified by the
Non-positive definite input parameter. The
following options are available:
•
Ignore – Proceed with the computation and do not issue an alert. The output
is not a valid factorization. A partial factorization will be present in the
upper left corner of the output.
SLL
*
=
91– 2
1– 85–
25– 7
3.00 0.33– 0.67
0.33– 2.81 1.70–
0.67 1.70– 1.91
L
3.00 0 0
0.33– 2.81 0
0.67 1.70– 1.91
=
- Blockset 1
- How to Contact The MathWorks: 2
- Contents 3
- Working with Signals 4
- DSP Function Reference 10
- Introduction 11
- 1 Introduction 12
- Welcome to the DSP Blockset 12
- What Is the DSP Blockset? 13
- Frame-Based Operations 13
- Matrix Support 14
- Statistical Operations 15
- Linear Algebra 15
- Parametric Estimation 15
- Real-Time Code Generation 15
- What Is in the DSP Blockset? 16
- Installation 17
- How to Use This Guide 19
- Signal Dimension Nomenclature 20
- Sampling Nomenclature 21
- R12 Related Products 23
- DSP Blockset 25
- The Simulink Environment 27
- Simulink on PC Platforms 27
- Simulink on UNIX Platforms 28
- The Simulink Libraries 29
- Model Definition 29
- parameters. They also both 31
- Model Simulation 32
- 0.1 for Time 33
- [1 5] or 33
- Learning More About Simulink 34
- Fixed-step option in the 35
- Using dspstartup.m 36
- Customizing dspstartup.m 36
- Performance-Related Settings 37
- Simulink and the DSP Blockset 38
- Parameters 38
- Miscellaneous Settings 39
- Signal Concepts 43
- Source Blocks 49
- Nonsource Blocks 50
- Error: Continuous sample 50
- Multichannel Signals 51
- Accelerating Simulations 55
- Sample Rates and Frame Rates 57
- 0, as explained in 58
- Red = fastest sample rate 59
- Direct Rate Conversion 61
- Sample time and Samples per 63
- Changing Frame Status 71
- Creating Signals 73
- •Sample time 77
- Constructing Signals 82
- Number of inputs = 2 85
- Signal = 5*ones(10,1) 87
- Initial 93
- Signal = randn(3,2,10) 95
- Select = Rows 95
- - Starting column index = 1 96
- 3 Working with Signals 100
- Deconstructing Signals 101
- Importing Signals 102
- Importing WAV Files 111
- Exporting Signals 112
- Using the To Workspace Block 116
- Stereo check box. See the 119
- Viewing Signals 120
- Example: Displaying Matrices 123
- = 0:pi/64:pi 124
- •Ending column offset = 1 124
- Delay and Latency 125
- Reducing Computational Delay 125
- Zero Algorithmic Delay 127
- Basic Algorithmic Delay 129
- Predicting Tasking Latency 132
- Type = Fixed-step 135
- Mode = SingleTasking 135
- DSP Operations 137
- 4 DSP Operations 138
- Overview 138
- Filters 139
- Multirate Filters 145
- Transforms 146
- Example: Using the FFT Block 146
- Example: Using the IFFT Block 148
- Power Spectrum Estimation 151
- Example: LU Solver 153
- Example: LU Inverse 155
- Statistics 157
- Example: Sliding Windows 158
- Running Operations 159
- DSP Blockset Demos Overview 160
- Communications Demos 161
- Filtering Demos 161
- DSP Block Reference 163
- 5 DSP Block Reference 164
- About Tunable Parameters 165
- About Supported Data Types 165
- Block Library List 171
- Analytic Signal IDCT 179
- Complex Cepstrum IFFT 179
- DCT Real Cepstrum 179
- Analog Filter Design 180
- Supported 183
- Data Types 183
- See Also 183
- Analytic Signal 184
- Autocorrelation 186
- Dialog Box 187
- Autocorrelation LPC 188
- Backward Substitution 192
- Biquadratic Filter 194
- Sample-Based Operation 198
- Sample-based input 199
- Frame-based output 199
- Frame-Based Operation 200
- Latency Zero Latency 201
- Nonzero Latency 202
- Burg AR Estimator 205
- Burg Method 207
- Check Signal Attributes 211
- Library DSP Sources 218
- These diagrams apply 220
- Initial frequency (Hertz), f 221
- Unidirectional 221
- Sweep time, T 221
- Bidirectional 221
- Sweep time, 2 221
- Block Computation Methods 222
- Quadratic, or Logarithmic 224
- 1 second), the Target 226
- Sweep time (1.5 seconds) 229
- Target frequency parameter 230
- AMPLITUDE 231
- Cholesky Factorization 235
- Cholesky Inverse 237
- Cholesky Solver 239
- Complex Cepstrum 241
- Complex Exponential 243
- Constant Diagonal Matrix 244
- parameter 246
- Constant Ramp 247
- Contiguous Copy 248
- Convert 1-D to 2-D 250
- Convert 2-D to 1-D 252
- ∗) as a guide to wiring 253
- ∗ symbol 253
- Convolution 257
- Correlation 259
- 5Counter 261
- Free-Running Operation 263
- Covariance AR Estimator 268
- Covariance Method 270
- Create Diagonal Matrix 272
- Cumulative Sum 273
- Power Inputs 275
- Voltage Inputs 275
- Library Transforms 279
- Delay Line 281
- 5Detrend 285
- Difference 286
- Digital Filter Design 288
- Initial Conditions 292
- Discrete Impulse 295
- Output Sequence (K=4) 298
- Downsample 299
- ) appears in the output as 303
- DSP Constant 305
- 5DSP Constant 305
- Dyadic Analysis Filter Bank 308
- Dyadic Synthesis Filter Bank 316
- Edge Detector 323
- Event-Count Comparator 325
- Extract Diagonal 327
- Extract Triangular Matrix 328
- Valid Block Inputs 331
- Dimension Along 331
- Which Block 331
- Computes DFT 331
- Twiddle factor computation 335
- Parameter Setting 335
- Sine and Cosine Computation 335
- Effect on Block Performance 335
- Speed 3N/4 1536 bytes 336
- Memory N/4 + 1 516 bytes 336
- Look under mask. Look under 337
- Filter Realization Wizard 340
- 5Filter Realization Wizard 340
- Fixed-Point Options 341
- Example 1: Direct Form II 342
- Example 3: Nth Order Sections 344
- Example 4: ARMA Lattice 345
- FIR Decimation 350
- Examples Example 1 353
- FIR Interpolation 357
- FIR Rate Conversion 364
- Forward Substitution 370
- Frame Status Conversion 372
- From Wave Device 374
- 5From Wave Device 374
- Buffering 375
- From Wave File 379
- Histogram 381
- Running Operation 382
- Identity Matrix 387
- Skip normaliza 391
- Conjugate Symmetric Input 392
- Inputs in Bit-Reversed Order 393
- Inherit Complexity 396
- Integer Delay 398
- Delay: [2 0 5] 404
- Initial conditions: 0 404
- Kalman Adaptive Filter 406
- LDL Factorization 411
- LDL Inverse 414
- LDL Solver 416
- Least Squares Polynomial Fit 418
- Levinson-Durbin 421
- Applications 422
- LMS Adaptive Filter 425
- LU Factorization 429
- LU Inverse 431
- LU Solver 432
- Magnitude FFT 434
- Matrix 1-Norm 437
- Matrix Multiply 439
- Matrix Product 441
- Matrix Scaling 443
- Matrix Square 445
- Matrix Sum 447
- Matrix Viewer 449
- Library Statistics 455
- Value Mode 455
- Index Mode 455
- Value and Index Mode 456
- Running Mode 456
- Basic Operation 460
- Modified Covariance Method 473
- Multiphase Clock 476
- Multiport Selector 479
- 5Multiport Selector 479
- N-Sample Enable 482
- N-Sample Switch 484
- 5N-Sample Switch 484
- Normalization 486
- Overlap-Add FFT Filter 488
- Overlap-Save FFT Filter 491
- Library Signal Operations 494
- Permute Matrix 496
- 5Permute Matrix 496
- Polynomial Evaluation 500
- Polynomial Stability Test 502
- Pseudoinverse 504
- QR Factorization 506
- QR Solver 508
- Register size parameter, and 510
- Clear input check box) 510
- Example 2 512
- Random Source 515
- Sample Period 518
- Real Cepstrum 522
- Reciprocal Condition 524
- Maintain input frame rate 527
- Initial condition 528
- RLS Adaptive Filter 532
- Sample and Hold 540
- Short-Time FFT 542
- Signal From Workspace 545
- Signal To Workspace 549
- Output Dimension Summary 550
- Sine Wave 555
- Singular Value Decomposition 562
- Spectrum Scope 566
- Standard Deviation 575
- Submatrix 579
- N/2 by the 581
- •Starting column offset = 1 582
- SVD Solver 587
- Time Scope 589
- 5Time Scope 589
- Library DSP Sinks 589
- Filter Type 590
- Filter Update Rate 592
- Time-Varying Lattice Filter 595
- Toeplitz 599
- 5Toeplitz 599
- To Wave Device 601
- 5To Wave Device 601
- To Wave File 606
- 5To Wave File 606
- Transpose 608
- Triggered Delay Line 610
- Trigger Event 613
- Initial and Final Conditions 613
- Triggered To Workspace 617
- 5Triggered To Workspace 617
- Unbuffer 620
- Uniform Decoder 623
- Uniform Encoder 627
- The Two Unwrap Modes 632
- Tolerance parameter) 634
- Tolerance parameter = π 634
- Unwrap Method 636
- Definition of Phase Unwrap 636
- Method” on page 5-479 637
- 0 pi 2pi 3pi 4pi 5pi 6pi 638
- Upsample 641
- Variable Fractional Delay 648
- Interpolation Modes 649
- Variable Integer Delay 653
- Variable Selector 661
- 5Variable Selector 661
- Variance 664
- Vector Scope 668
- Display Properties 672
- Axis Properties 673
- Line Properties 675
- Scope Window 678
- Display Properties Dialog Box 681
- Line Properties Dialog Box 684
- Wavelet Analysis 686
- Filter Coefficients 687
- Tree Structure 687
- Wavelet Synthesis 692
- Window Function 698
- Window Sampling 699
- Window Type 699
- Yule-Walker AR Estimator 703
- Yule-Walker Method 706
- Zero Pad 709
- 6 DSP Function Reference 713
- Syntax dsp_links 714
- Syntax dsplib 715
- 6dspstartup 716
- Syntax dspstartup 716
- Syntax liblinks 718
- Numerics 720
- RTWOptions parameter 2-15 733
- SaveOutput parameter 2-13 733
- SaveTime parameter 2-13 733
- Solver parameter 2-15 735
- StopTime parameter 2-15 735
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