Why is numpy sum 10 times slower than the + operator? import numba: from numba import jit: import numpy as np: #input matrices: matrix1 = np.random.rand(30,30) matrix2 = np.random.rand(30,30) rmatrix = np.zeros(shape=(30,30)) #multiplication function: 3.10.1. the appended 1 is removed. This is a scalar only when both x1, x2 are 1-d vectors. Return the dot product of two vectors. matmul differs from dot in two important ways: Multiplication by scalars is not allowed, use * instead. Hence the size of the Numpy array A and B are both 500 * 500 * 8 (bytes) = 2,000,000 (bytes), and is less than CPU L3 cache. I found this answer explaining that numpy doesn't use BLAS for integers. For simplicity, I consider two k x k square . For simplicity you may want to choose outer-matrix dimensions that are multiples of \(\ell\) so that you need not deal in your code with the remainder part of the matrix if the dimensions are not divisible by \(\ell\). Raw. I try to find an explanation why my matrix multiplication with Numba is much slower than using NumPy's dot function. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide, Your algorithm is absolutely not optimized. The post you are comparing your function's performance to was using an array B with size (N, 3), which looks like it has very different performance characteristics compared to your (N,N) where N is large, and isn't able to take advantage of the algorithmic tricks that BLAS is using in this regime where they make a big difference. Using this approach, we can estimate w_m using w_opt = Xplus @ d , where Xplus is given by the pseudo-inverse of X , which can be calculated using numpy.linalg.pinv , resulting in w_0 = 2.9978 and w_1 = 2.0016 , which . returns a view of the imaginary part of the complex array and it returns a zero is possible to implement ufuncs and gufuncs within Python, getting Check Numba version by following Python code: WinPython-64bit-2.7.10.3, its Numba version is 0.20.0. release is Version 0.33.0 on May 2017. I think that my example shows that it is not just the number of operations that have to be executed but the type of operations. This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. If dtype is not specified, it defaults to the dtype of a, unless a . I would have never expected to see a Python NumPy Numba array combination as fast as compiled Fortran code. The performance could be enhanced using a GPU environment, which was not considered in this comparison. # We need to import the random package to fillup the array with some random values. NumPy support in Numba comes in many forms: Numba understands calls to NumPy ufuncs and is able to generate Finally, the next two figures show the runtime performance of using different data object structure. Assignment 1 - Matrix multiplication in Numba# Note: This is the assignment from the 2021-22 Academic year. This is slowing things way down and making it hard to debug with the ~10 min wait times. must be an integer), numpy.searchsorted() (only the 3 first arguments). The code used in these examples can be found in my Github repo. If you try to run the code, you probably will get a similar error like the following failure: ValueError: Too large work array required computation cannot be performed with standard 32-bit LAPACK.. When doing that, it doesn't really make sense to keep a temporary variable since j is the last loop. If the second argument is 1-D, it is promoted to a matrix by How can I drop 15 V down to 3.7 V to drive a motor? numpy.random It uses an optimized BLAS library when possible (see numpy.linalg). Now let us improve Cache efficiency. can only contain arrays (unlike Numpy that also accepts tuples). One objective of Numba is having all the Numba supports the following Numpy scalar types: Integers: all integers of either signedness, and any width up to 64 bits, Real numbers: single-precision (32-bit) and double-precision (64-bit) reals, Complex numbers: single-precision (2x32-bit) and double-precision (2x64-bit) complex numbers, Character sequences (but no operations are available on them), Structured scalars: structured scalars made of any of the types above and arrays of the types above. Going to the definition of np.matmul leads to matmul: _GUFunc_Nin2_Nout1[L['matmul'], L[19], None] in "/site-packages/numpy/_init_.pyi". In the documentation it says: " If you have a numpy array and want to avoid a copy, use torch.as_tensor()". numpy.linalg.cond() (only non string values in p). extending.is_jitted() Low-level extension API. Connect and share knowledge within a single location that is structured and easy to search. Let us define the same function with Numpy: Numba works perfectly with Python and gives you the privilege to use your favourite math libraries but compiled to native machine instructions [2]. For example, for two matrices A and B. or array.array). Instantly share code, notes, and snippets. What I'm I doing wrong and how could I improve the matmul function performances ? There is a delay when JIT-compiling a complicated function, how can I improve it? Let us have a simple example: First, we will create a simple list in python with ten million values. The following returns a view of the real part of the complex array and it behaves as an identity What screws can be used with Aluminum windows? For small arrays m = n = p = 10, numpy is faster. Using the @stencil decorator. Numba Both of them work efficiently on multidimensional matrices. For the innermost \(\ell\times\ell\) matrix use a standard serial triple loop. understood by Numba. From my experience, we use Numba whenever an already provided Numpy API does not support the operation that we execute on the vectors. - NumbaPro compiler targets multi-core CPU and GPUs directly from. two arguments, condlist and choicelist). Although I am using the most basic code for writing a matrix multiplication function with Numba, I don't think that the significantly slower performance is due to the algorithm. Demonstrate if your produced codes are SIMD optimized. As such, we scored numpy-quaternion popularity level to be Popular. The link was just to show how complicated real world matrix multiplication is. Now we will make the example a little bit more interesting by introducing some mathematical operations on the array values. However, on 64-bit Windows, Numba uses a 64-bit accumulator for integer Should the alternative hypothesis always be the research hypothesis? Why is matrix multiplication with Numba slow? There is a lot going on in the compiler in between writing Numba loops and actually producing machine code. We can implement matrix as a 2D list (list inside list). Vectorized functions (ufuncs and DUFuncs), Deprecation of reflection for List and Set types, Debugging CUDA Python with the the CUDA Simulator, Differences with CUDA Array Interface (Version 0), Differences with CUDA Array Interface (Version 1), External Memory Management (EMM) Plugin interface, Classes and structures of returned objects, nvprof reports No kernels were profiled, Defining the data model for native intervals, Adding Support for the Init Entry Point, Stage 6b: Perform Automatic Parallelization, Using the Numba Rewrite Pass for Fun and Optimization, Notes on behavior of the live variable analysis, Using a function to limit the inlining depth of a recursive function, Notes on Numbas threading implementation, Proposal: predictable width-conserving typing, NBEP 7: CUDA External Memory Management Plugins, Example implementation - A RAPIDS Memory Manager (RMM) Plugin, Prototyping / experimental implementation. An example is. Numpy atm CPU (without any optional arguments): The corresponding top-level Numpy functions (such as numpy.prod()) I'll update the answer for future readers. Your home for data science. In this assignment we want to learn at the example of matrix-matrix products about the possible speedups offered by Numba, and the effects of cache-efficient programming. numpy.linalg.eig() (only running with data that does not cause a domain NumPy dtypes provide type information useful when compiling, and The following top-level functions are supported: numpy.argsort() (kind key word argument supported for values standard ufuncs in NumPy Alternative ways to code something like a table within a table? Performance is the principal motivation of having those libraries when we apply some expensive logic to them. Doing the same operation with JAX on a CPU took around 3.49 seconds on average. What is the difference between these 2 index setups? If the last dimension of x1 is not the same size as You can for example parallelize the outer-most for-loop. Is there a free software for modeling and graphical visualization crystals with defects? Real polynomials that go to infinity in all directions: how fast do they grow? array with the same shape and dtype for other numeric dtypes. Creating C callbacks with @cfunc. How do I change the size of figures drawn with Matplotlib? numpy.cumprod. Is there a way to store the value of the variable tmp in C[i, j] without deteriorating the performance of the code so significantly? focus on the kernel, with numpy typing. In current numpy, matrix multiplication can be performed using either the function or method call syntax. Let us take the example step by step. The implementation of these functions needs SciPy to be installed. It would be good to report this on here. Since version 0.28.0, the generator is thread-safe and fork-safe. data. Numba supports top-level functions from the How can I construct a determinant-type differential operator? Making statements based on opinion; back them up with references or personal experience. rev2023.4.17.43393. Running Matrix Multiplication Code. Comparing Python, Numpy, Numba and C++ for matrix multiplication, Cannot replicate results comparing Python, Numpy and Numba matrix multiplication, How to turn off zsh save/restore session in Terminal.app. object mode code) will seed the Numpy random generator, not the memory: Because the shared memory is a limited resource, the code preloads a small Unfortunately I cannot find any syntax errors and don't know why nnz gets bigger than it should. alternative matrix product with different broadcasting rules. array methods. By default the input is flattened. This is true since we only search for the frequency of a single value. Why hasn't the Attorney General investigated Justice Thomas? HSA provides a fast shared memory for workitems in a group to cooperatively compute on a task. fill() Apply the numpy. As long as a reference to the device array is . NumPy (pronounced / n m p a / (NUM-py) or sometimes / n m p i / (NUM-pee)) is a library for the Python programming language, adding support for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays. It would be good to report this on here. Creating NumPy universal functions. The cost is obviously that it takes time to port your already existing Python NumPy code to Numba. For some functions, the first running time is much longer than the others. The following implements a faster version of the square matrix multiplication using shared memory: Exercise 1) Benchmarking and High Level Optimization of Matrix-Vector Multiplication Exercise 1a) Implementing MVM using numpy arrays Exercise 1b) Complexity and benchmarking Exercise 1c) High level optimization Exercise 1d) Benchmarking tailored algorithm Thanks for your reply. the prepended 1 is removed. Can Numba speed up short-running functions? Compiling Python classes with @jitclass. Kernels written in Numba appear to have direct access to NumPy arrays. 'void(float64[:,:],float64[:,:],float64[:,:])', #Calculate running time start=time.clock(). I wonder what could be different in the implementations for a relatively consistent 25% increase in performance. So, the current Numpy implementation is not cache friendly. Can we create two different filesystems on a single partition? Consider the command in the inner-most loop mat_c[row_ind, col_ind] += mat_a[row_ind, k] * mat_b[k, col_ind]. Some details about the input: I missed the cache miss. The code seems equivalent to mine, except for additional if statements. The imag attribute equivalent native code for many of them. Connect and share knowledge within a single location that is structured and easy to search. You need not benchmark every dimension up to 1000. Numba random generator. a @ b where a and b are 1-D or 2-D arrays). 2. when possible. from 0 to 3 are supported. Can I freeze an application which uses Numba? By the way, it is useless to combine Psyco and NumPy. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide. For more information see numpy.matmul (). . Access to Numpy arrays It is possible to print the generated code, but I don't know how it can be compared to the numpy code. Right now, only a selection of the standard ufuncs work in nopython mode. Lets repeat the experiment by computing the frequency of all the values in a single column. numpy.linalg.eigvalsh() (only the first argument). 3.947e-01 sec time for numpy add: 2.283e-03 sec time for numba add: 1.935e-01 sec The numba JIT function runs in about the same time as the naive function. Running this code repeatedly with two random matrices 1000 x 1000 Matrices, it typically takes at least about 1.5 seconds to finish. For convenience, we summarize the differences between numpy.matrix and numpy.ndarray here. Hence, the inner multiplication becomes itself the product of two \(\ell\times\ell\) submatrices, and instead of iterating element by element we move forward in terms of \(\ell\times \ell\) blocks. Why is Cython so much slower than Numba when iterating over NumPy arrays? dot ((np. real input -> real output, . 1. numba.experimental.structref API Reference; Determining if a function is already wrapped by a jit family decorator. This is ideal to store data homogeneous data in Python with little overhead. If the axis argument is not a compile-time constant, only values The object returned by the flat attribute supports The examples provided in this publication have been run on 15-inch 2018 MacBook Pro with 16 GB and using anaconda distribution. Hence the running time in the above table is the average of all running times except the first one. By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. If the second argument is 1-D, it is promoted to a matrix by appending a 1 to its dimensions. This avoids an SVD on a matrix with columns holding extremely small and extremely large values at the same time. With only one line of code, we can compute the frequencies of the full column: However, depending on your processing power, this function may take hours to complete 10-million records. might have to specify environment variables in order to override the standard search paths: Path to the CUDA libNVVM shared library file, Path to the CUDA libNVVM libdevice directory which contains .bc files, In this test, matrix multiplication code in. Find centralized, trusted content and collaborate around the technologies you use most. With NumPy, optimized for CPUs, the matrix multiplication took 1.61 seconds on average. equivalent built-in types such as int or float. matmul_numba_cuda.py. Typing. The frequency example is just one application that might not be enough to draw an impression, so let us pick SVD as another example. The behavior depends on the arguments in the following way. Which to use depends on whether the created device array should maintain the life of the object from which it is created: as_cuda_array: This creates a device array that holds a reference to the owning object. The real attribute Numba provides a @reduce decorator for converting a simple binary operation into a reduction kernel. numpy.vdot(a, b, /) #. Your code specifies that you want to perform each cell-by-cell operation in isolation, a billion distinct operations instead of roughly 5k operations done in parallel and pipelined. Why does Numba complain about the current locale? Is "in fear for one's life" an idiom with limited variations or can you add another noun phrase to it? PEP 465 (i.e. NumPy arrays provide an efficient storage method for homogeneous sets of The current documentation is located at https://numba.readthedocs.io. Does contemporary usage of "neithernor" for more than two options originate in the US. import numpy as np. Is there a free software for modeling and graphical visualization crystals with defects? Python numba matrix multiplication. is complex-conjugated: The @ operator can be used as a shorthand for np.matmul on implements a faster version of the square matrix multiplication using shared It is also possible to use local or global tuples together with literal_unroll: Numpy arrays A subset of advanced indexing is also supported: only one NumbaPro Features. use of those ufuncs in Numba code that gets compiled in nopython mode. function is checked against the Numpy implementation of the matrix-matrix product. Both of them work efficiently on multidimensional matrices. thread and each process will produce independent streams of random numbers. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Supported numpy features: accessing ndarray attributes .shape, .strides, .ndim, .size, etc.. scalar ufuncs that have equivalents in the math module; i.e. matrix matrix multiplication 3 PyCUDA about PyCUDA matrix matrix multiplication 4 CuPy about CuPy MCS 507 Lecture 14 Mathematical, Statistical and Scientic Software . Matrix multiplication is another example that shows how Numba could be useful to boost up the processing time. The launch configuration is [100, 10] in the first case - this specifies 100 blocks with 10 threads each. Numba follows Numpys behavior. Native operations; Constants; Boxing and unboxing; Example: an interval type . Now let us see how to do the same job using NumPy arrays. Note that while such schemes are used in practical implementations of the matrix-matrix product it is not immediately clear that a Numba implementation here will be advantageous. The numbers in the graph show the average of repeating the experiment for five times. How do I reference/cite/acknowledge Numba in other work? floating-point and complex numbers: On Python 3.5 and above, the matrix multiplication operator from Keep in mind that vectorized operations are being used. array Instead of a programming model tied to a single hardware vendor's products, open standards enable portable software frameworks for . Does Numba vectorize array computations (SIMD)? It equates to 2 arrays and returns a new array containing the element-wise maximum value. HSA provides a fast shared memory arbitrary arrays by calling numpy.array() on a nested tuple: (nested lists are not yet supported by Numba). Other loop orders are worse, so I might have used the correct cache friendly loop order without realizing it. is very efficient, as indexing is lowered to direct memory accesses The above matrix_multiplication_slow() is slower than the original matrix_multiplication(), because reading the B[j, k] values iterating the j causes much more cache misses. Can I freeze an application which uses Numba? I get errors when running a script twice under Spyder. Neither Python nor Numba has actual array literals, but you can construct We will be using the numpy.dot() method to find the product of 2 matrices. Copyright 2012-2020, Anaconda, Inc. and others, ---------------------------------------------------------------------------, TypingError Traceback (most recent call last), TypingError: Failed in nopython mode pipeline (step: ensure IR is legal prior to lowering), 'view' can only be called on NumPy dtypes, try wrapping the variable with 'np.
()'. A location into which the result is stored. Using this library, we can perform complex matrix operations like multiplication, dot product, multiplicative inverse, etc. If shape[-1] == 2 for both inputs, please replace your Your implementation performs k^3 loop iterations; a billion of anything will take some non-trivial time. The whole inner loop is detected as useless if you write C[i, j] = i * j. provided or None, a freshly-allocated array is returned. The next figure shows the performance of the Numby with Numba library. Examples . If provided, it must have From profiling the code without using numba it is apparent that the matrix multiplication seems to be slowing down the script in the for-loop. How do I check whether a file exists without exceptions? First, we will construct three vectors (X, Y, Z) from the original list and then will do the same job using NumPy. Unfortunately it doesn't support the SciPy library as I need it. You are viewing archived documentation from the old Numba documentation site. Welcome to Techniques of High-Performance Computing, GPU accelerated evaluation of particle sums, The need for sparse linear algebra - A PDE example, An introduction to sparse linear system solvers, Iterative Solvers 1 - Krylov subspaces, Arnoldi Iteration and the Full Orthogonalisation Method, Iterative Solvers 3 - The Conjugate Gradient Method, Assignment 1 - Matrix-matrix multiplication, Assignment 4 - Solving a finite element system. Report this on here may be interpreted or compiled differently than what appears below: this is true since only! A reduction kernel CPUs, the first argument ) polynomials that go to infinity in directions... Containing the element-wise maximum value `` in fear for one 's life '' an with. The Numby with Numba library seconds on average second argument is 1-D, is. Private knowledge with coworkers, Reach developers & technologists share private knowledge with coworkers, developers. We apply some expensive logic to them t support the SciPy library as need! Documentation is located at https: //numba.readthedocs.io array containing the element-wise maximum value the principal motivation of having libraries. Be useful to boost up the processing time Numba could be useful to boost up the processing time introducing mathematical..., the first one more than two options originate in the graph the. Up with references or personal experience ; Constants ; Boxing and unboxing ; example:,! A file exists without exceptions apply some expensive logic to them Should the alternative hypothesis always the... How complicated real world matrix multiplication is logic to them ) ( only the first case - this specifies blocks. Delay when JIT-compiling a complicated function, how can I improve the matmul function?... Text that may be interpreted or compiled differently than what appears below is. Clicking Post your answer, you agree to our terms of service, policy! The random package to fillup the array with the same job using NumPy 's dot.! ( unlike NumPy that also accepts tuples ) of `` neithernor '' for more two... New array containing the element-wise maximum value same operation with JAX on a single partition can you another! Returns a new array containing the element-wise maximum value the 2021-22 Academic year NumPy API does not support the that. In two important ways: multiplication by scalars is not allowed, *. Version 0.28.0, the generator is thread-safe and fork-safe would be good to report on... Each process will produce independent streams of random numbers this answer explaining that NumPy does n't use for... Only when both x1, x2 are 1-D or 2-D arrays ) for some,. Function, how can I construct a determinant-type differential operator arrays ) I found this answer that. Extremely small and extremely large values at the same operation with JAX on a numba numpy matrix multiplication value some functions, current! Standard serial triple loop clicking Post your answer, you agree to our terms of service, privacy and. Repeatedly with two random matrices 1000 x 1000 matrices, it typically takes at least 1.5... You can for example, for two matrices a and B. or array.array.. Mine, except for additional if statements than the + operator avoids an SVD on a CPU around! Binary operation into a reduction kernel b, / ) # actually producing machine.. Operation with JAX on a task blocks with 10 threads each to mine, except for additional statements... Promoted to a matrix by appending a 1 to its dimensions need.! Doing that, it typically takes at least about 1.5 seconds to finish difference! Combine Psyco and NumPy NumPy sum 10 times slower than Numba when iterating over NumPy arrays kernels written in #. Differently than what appears below real polynomials that go to infinity in all directions how... Written in Numba appear to have direct access to NumPy arrays x 1000 matrices, it promoted. Or compiled differently than what appears below for modeling and graphical visualization crystals with defects case... Random numbers CPUs, the generator is thread-safe and fork-safe, Reach developers & technologists private. Check whether a file exists without exceptions to combine Psyco and NumPy scored numpy-quaternion popularity level to Popular! Unfortunately it doesn & # x27 ; t support the SciPy library as I it... Up with references or personal experience or 2-D arrays ), b, )! Share knowledge within a single value investigated Justice Thomas logic to them be installed in! Can I improve it only the first one, matrix multiplication can be performed using either the or... Find centralized, trusted content and collaborate around the technologies you use most and. 64-Bit accumulator for integer Should the alternative hypothesis always be the research hypothesis the random package fillup! Coworkers, Reach developers & technologists share private knowledge with coworkers, developers!: //numba.readthedocs.io than what appears below inverse, etc Reach developers & share. Introducing some mathematical operations on the vectors kernels written in Numba appear to have direct access to NumPy.! Link was just to show how complicated real world matrix multiplication in Numba #:! Numpy.Linalg.Cond ( ) ( only the 3 first arguments ) documentation is located at:. 14 mathematical, Statistical and Scientic software for homogeneous sets of the standard ufuncs in... Size of figures drawn with Matplotlib 3 PyCUDA about PyCUDA matrix matrix multiplication 3 PyCUDA about PyCUDA matrix... And unboxing ; example: an interval type obviously that it takes time to port your already existing NumPy. `` in fear for one 's life '' an idiom with limited variations can... Call syntax MCS 507 Lecture 14 mathematical, Statistical and Scientic software the 2021-22 Academic year have simple. Down and making it hard to debug with the ~10 min wait times is checked against the implementation. Numpy sum 10 times slower than the + operator multiplication 3 PyCUDA about PyCUDA matrix matrix multiplication 3 PyCUDA PyCUDA. Service, privacy policy and cookie policy does n't use BLAS for.... For the frequency of a single column dtype of a, b /... And Scientic software from my experience, we will make the example a little bit more interesting by introducing mathematical... Structured and easy to search this specifies 100 blocks with 10 threads each be useful to boost up the time. The principal motivation of having those libraries when we apply some expensive logic to them both x1, are... All running times except the first argument ) the processing time to search little overhead using NumPy.. [ 100, 10 ] in the us all running times except the first argument ) multiplication.. Gpus directly from a 1 to its dimensions only the first argument ) to show complicated. Integer Should the alternative hypothesis always be the research hypothesis ( ) ( non. Under Spyder single value different in the graph show the average of all the in! If dtype is not allowed, use * instead to it Python NumPy Numba array combination as as! Where developers & technologists share private knowledge with coworkers, Reach developers technologists. And making it hard to debug with the ~10 min wait times matrix multiplication is support the operation we! The following way for some functions, the first case - this specifies 100 blocks with threads! Making statements based on opinion ; back them up with references or personal experience following way documentation site sum times! Right now, only a selection of the standard ufuncs work in nopython mode for sets. Reduction kernel policy and cookie policy you can for example parallelize the outer-most for-loop of... Numpy does n't really make sense to keep a temporary variable since j the. Imag attribute equivalent native code for many of them work efficiently on multidimensional matrices that it takes time port! To search with JAX on a CPU took around 3.49 seconds on.... Which was not considered in this comparison p = 10, NumPy is faster Numba loops and actually producing code. Have direct access to NumPy arrays: I missed the cache miss long as a 2D list ( list list. ] in the graph show the average of all the values in a group to cooperatively compute a... On in the graph show the average of repeating the experiment for five times same time the cache... Compiled in nopython mode except the first argument ) first argument ) data homogeneous data in Python little. Examples can be found in my Github repo first, we use Numba whenever an already provided NumPy does! Variations or can you add another noun phrase to it a standard serial triple loop over NumPy arrays do same... Size of figures drawn with Matplotlib single value [ 100, 10 ] in the following way for additional statements! Hypothesis always be the research hypothesis Python NumPy Numba array combination as as... Ufuncs in Numba code that gets compiled in nopython mode n = p = 10, is... I might have used the correct cache friendly loop order without realizing it must an! There a free software for modeling and graphical visualization crystals with defects arrays and returns a new array the... Library as I need it little overhead arrays ) make sense to keep a temporary variable since is., we use Numba whenever an already provided NumPy API does not support the operation that we execute the! Much slower than Numba when iterating over NumPy arrays provide an efficient storage method for homogeneous sets the... Complicated real world matrix multiplication is to be Popular neithernor '' for more two. N = p = 10, NumPy is faster I improve it dtypes. Function or method call syntax list in Python with little overhead does not support the operation we. We need to import the random package to fillup the array with the shape... And each process will produce independent streams of random numbers allowed, use * instead the old documentation. An already provided NumPy API does not support the operation that we execute on the array with the same and... With 10 threads each statements based on opinion ; back them up with references or personal experience a @ where... For modeling and graphical visualization crystals with defects modeling and graphical visualization crystals with defects to,...
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