Sharp JX-9400 Technické informace Strana 184
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- KNIHY
Hodnocené. / 5. Na základě hodnocení zákazníků
and the matrix norm is subordinated to the vectorial norm jxj if:
jAj¼max
jAxj
jxj
for any x 6¼ 0 ð7:85Þ
The subordinated norm is the smallest matrix norm compatible with the
norm jxj.
For example, the norm jAj
2
, defined as:
jAj
2
¼
ffiffiffiffiffi
1
p
ð7:86Þ
where
1
is the largest eigenvalue of A
H
A (A
H
¼ hermitic conjugate or trans-
pose of the complex conjugate matrix) is subordinated to the Euclidian norm
jxj
2
but the Frobisher norm:
jAj
F
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
i
a
2
i
X
j
a
2
j
s
ð7:87Þ
is consistent with the Euclidian norm but not subordinated to it.
Finally, the following norms, which lead to faster calculations, are often
used:
jxj
1
¼
X
i
jx
i
jð7:88Þ
and the corr esponding norm for the matrix A:
jAj
1
¼ max jA
j
jð7:89Þ
where A
j
are the column vectors of A.
Calculation of the upper bound
From the norms of the experimental errors y and A, it is possible to calculate
an upper limit to the norm of the resulting error x by the use of the following
relation.
If j Ij¼1 (it is true for jIj
2
), then the norm of the relative error is:
jxj
jxj
jAj jA
1
j
1 jAj jA
1
j
jyj
jyj
þ
jAj
jAj
ð7:90Þ
The quantity:
condðAÞ¼jAj jA
1
jð7:91Þ
is of great importance in this calculation. It is the so-called condition number of
the matrix A related to the used norm. This number indicates how nearly
singular the matrix is.
If the spectral norm jAj
2
is used, we get the smallest possible condition
number, which is:
cond
2
ðAÞ¼jAj
2
jA
1
j
2
¼
ffiffiffiffiffiffiffiffiffiffi
1
n
p
ð7:92Þ
Common Methods and Techniques 163
- Claude-Alain Roulet 4
- Contents 6
- List of Figures and Tables 8
- List of Figures and Tables ix 10
- List of Figures and Tables xi 12
- Preamble 13
- Introduction 14
- Outdoor airflow rate 15
- Recirculation rates 16
- Exfiltration 17
- Ventilation efficiency 18
- Airtightness 20
- Energy efficiency 20
- Common techniques 21
- Airflow Rates in Buildings 22
- Tracer decay, no injection 24
- Constant injection rate 24
- Constant concentration 25
- Pulse injection 25
- Airflow Rates in Buildings 5 26
- Airflow Rates in Buildings 7 28
- Global system of equations 29
- Airflow Rates in Buildings 9 30
- Properties of the flow matrix 31
- Airflow Rates in Buildings 11 32
- Note: y Under condition 34
- Source: Sherman, 1990 34
- Airflow Rates in Buildings 13 34
- Airflow Rates 36
- Source: Roulet et al., 2000a 37
- Velocity traverse 38
- Test procedure 39
- Tracer gas dilution 39
- Source: Awbi, 2007 40
- Inflatable bag 41
- Flowmeter 41
- Compensated flowmeter 41
- Tracer gas injection ports 41
- Source: Roulet et al., 1999 42
- Node by node method 44
- Least square solution 48
- Eliminating some equations 48
- Simplest way 49
- Less than four tracer gases 53
- Planning tool 54
- Simple measurement using CO 55
- Age of Air and 60
- Ventilation Efficiency 60
- Nominal time constant 61
- Air exchange efficiency 61
- Source: Roulet, 2004 62
- Measurement method 63
- Concentration [ppm] 65
- Source: Roulet et al., 1998 69
- The model matrix M 73
- Condition of the model matrix 74
- Factorial designs 75
- 3-D designs 76
- Measurement methods 80
- Reductive sealing 82
- Density corrections 84
- Two measurement points 85
- Airtightness 65 86
- Airtightness of buildings 88
- External fan 89
- Internal fan 90
- Leakage visualization 90
- The stack effect method 91
- Airtightness 71 92
- Neutral height method 93
- Airtightness 73 94
- Pressurization method 95
- Flow rate difference method 96
- Conditioned space 97
- SealSeal 97
- Measurements and Measures 98
- Related to Energy Efficiency 98
- Energy in air handling units 100
- Heat exchangers 104
- Types of heat exchangers 105
- Rotating heat exchangers 106
- Glycol heat exchanger 107
- Heat pump 107
- Heat exchange efficiency 107
- Ventilated 110
- Global heat recovery efficiency 111
- Source: Roulet et al., 2001 114
- Examples of application 115
- Energy for ventilation 118
- Measurement of airflow rate 120
- Measurement of electric power 121
- Simulation results 126
- Recirculation 126
- Humidification 127
- Heat recovery 127
- Ductwork 128
- Contaminants in 129
- Air Handling Units 129
- Source: Bjo 130
- Humidifiers 131
- Source: Roulet et al., 2000 133
- Measurement protocols 134
- Selecting the panel 135
- Source: Bluyssen, 1990 136
- Training procedure 137
- Experimental day 138
- Principle of the method 138
- Injection technique 139
- Hot air blower (200 °C) 140
- Stainless steel 140
- Ø 6 mm copper tube 140
- Example of application 141
- Evidence for adsorption 144
- HVAC systems 146
- Common Methods 153
- Properties of tracer gases 155
- Source: Dietz et al., 1983 157
- Mixing tracer gases 158
- Tracer density problem 159
- Sampling methods 160
- Grab sampling 160
- Passive sampling 160
- Networks, pumps and pipes 161
- Tracer gas analysers 163
- Objective of the analysis 163
- Photo-acoustic detector 165
- Mass spectrometry 165
- Gas chromatography 167
- Chemical indicator tubes 167
- Identification methods 168
- Confidence in the coefficients 170
- Regression of the second kind 171
- Orthogonal regression 171
- Bayesian identification 172
- Error analysis 174
- Definitions 175
- A few statistics 175
- Covariance 177
- Statistical distributions 177
- What is the problem? 180
- Most simple error analysis 180
- Estimate of the variance 181
- Linear equations systems 181
- Upper bound of the errors 183
- References 187
- Unit Conversion Tables 192
- Pressure 194
- Volume flow rate 194
- Mass flow rate 194
- Glossary 195
- Glossary 175 196
- Glossary 177 198
- Glossary 179 200
- Glossary 181 202
- Glossary 183 204
- Glossary 185 206
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