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documentation:language_reference:objects:responsefunction:start [2024/12/18 16:43] – Maurits W. Haverkortdocumentation:language_reference:objects:responsefunction:start [2025/11/20 04:15] (current) – external edit 127.0.0.1
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 ====== Response function ====== ====== Response function ======
  
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 Internally response functions can be stored in different formats. We need several formats as (1) transformations between the different formats take time (2) transformations between different formats can involve a loss of numerical accuracy and (3) different algorithms require the response function in different formats. The formats used in Quanty to store response functions are Internally response functions can be stored in different formats. We need several formats as (1) transformations between the different formats take time (2) transformations between different formats can involve a loss of numerical accuracy and (3) different algorithms require the response function in different formats. The formats used in Quanty to store response functions are
   - List of poles $$ G(\omega,\Gamma) = A_0 + \sum_{i=1}^{n} B_{i-1} \frac{1}{\omega + \mathrm{i}\Gamma/2 - a_i} $$   - List of poles $$ G(\omega,\Gamma) = A_0 + \sum_{i=1}^{n} B_{i-1} \frac{1}{\omega + \mathrm{i}\Gamma/2 - a_i} $$
-  - Tri-diagonal $$ G(\omega,\Gamma) = A_0 + B_0^* \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_1 - B_{1}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_2 - B_{2}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_2 - B_{3}^{\phantom{\dagger}} \frac{...}{\omega + \mathrm{i}\Gamma/2 - A_{n-1} - B_{n-1}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_n } B_{n-1}^{\dagger}} B_{3}^{\dagger} } B_{2}^{\dagger} } B_{1}^{\dagger} } B_0^T $$+  - Tri-diagonal $$ G(\omega,\Gamma) = A_0 + B_0^* \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_1 - B_{1}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_2 - B_{2}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_3 - B_{3}^{\phantom{\dagger}} \frac{...}{\omega + \mathrm{i}\Gamma/2 - A_{n-1} - B_{n-1}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_n } B_{n-1}^{\dagger}} B_{3}^{\dagger} } B_{2}^{\dagger} } B_{1}^{\dagger} } B_0^T $$
   - Anderson $$ G(\omega,\Gamma) = A_0 + B_0^* \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_1 - \sum_{i=2}^{n} B_{i-1}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_{i} } B_{i-1}^{\dagger} } B_0^T $$   - Anderson $$ G(\omega,\Gamma) = A_0 + B_0^* \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_1 - \sum_{i=2}^{n} B_{i-1}^{\phantom{\dagger}} \frac{1}{\omega + \mathrm{i}\Gamma/2 - A_{i} } B_{i-1}^{\dagger} } B_0^T $$
   - Natural Impurity. We define $G_{val}(\omega,\Gamma)$ and $G_{con}(\omega,\Gamma)$ as response functions with poles either at positive energy ($G_{con}(\omega,\Gamma)$) or poles at negative energy ($G_{val}(\omega,\Gamma)$). The full response function is defined as $$ G(\omega,\Gamma) = A_0 + B_0^* \left( G_{val}(\omega,\Gamma) + G_{con}(\omega,\Gamma) \right) B_0^T$$    - Natural Impurity. We define $G_{val}(\omega,\Gamma)$ and $G_{con}(\omega,\Gamma)$ as response functions with poles either at positive energy ($G_{con}(\omega,\Gamma)$) or poles at negative energy ($G_{val}(\omega,\Gamma)$). The full response function is defined as $$ G(\omega,\Gamma) = A_0 + B_0^* \left( G_{val}(\omega,\Gamma) + G_{con}(\omega,\Gamma) \right) B_0^T$$ 
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 Below you find several functions, methods, properties and operations related to response functions. Below you find several functions, methods, properties and operations related to response functions.
  
-Response functions can be stored internally as Lua tables with the a meta table set, or as a user data. Both objects behave similarly. One should be careful as a comparison between a response function stored as a table to a response function stored as a user data is always falls. That said, comparing functions of doubles will always be problematic due to numerical inaccuracies.+Response functions can be stored internally as Lua tables with a metatable set to ResponseFunctionMeta, or as a user data. Both objects behave similarly. One should be careful as a comparison between a response function stored as a table to a response function stored as a user data is always falls. That said, comparing functions of doubles will always be problematic due to numerical inaccuracies.
  
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