Sparse Matrix To Numpy Array

array is not the same as the Standard Python Library class array. Web Design Given an arbitrary numpy array (ndarray), is there a function or a short way to convert it to a scipy. I have a numpy array with m columns and n rows, the columns being dimensions and the rows datapoints. Source Code: Matrix Multiplication. This NumPy stack has similar users to other applications such as MATLAB , GNU Octave , and Scilab. numpy,sparse-matrix,sparse,sparse-array. If you rely on the fact that there is at most a constant number of non-zero elements in a single column, it is your responsibility to ensure that the matrix is symmetric. Whether transform should produce scipy. the multiplication with '*' is the matrix multiplication (dot product); not part of NumPy! passing a sparse matrix object to NumPy functions expecting ndarray/matrix does not work. just what you need. Let us first create some data in (i,j,v) format. TypeError: Singleton array array(<4020784x50626 sparse matrix of type '' with 151426374 stored elements in Compressed Sparse Row format>, dtype=object) cannot be considered a valid collection. All arguments (even the positional) are optional. mount of time to allocate this memory. basics of numpy , sparse matrix Tools Needed Anaconda , Jupyter Notebook, python 3. sparse matrix? <. A can be a scipy sparse matrix or a numpy array. create numpy arrays, slice arrays, merge arrays, basic types of numpy arrays, reading and writing arrays to file, reading and writing sparse matrices to svmlight format. When we create a matrix, we generally know what type of data will be stored in the matrix, how many dimensions it will have and how many elements. This implementation will refuse to center scipy. True by default. They are from open source Python projects. SciPy创建稀疏矩阵. Separator string used when constructing new features for one-hot coding. Delete given row or column. so if matrix looks this. Calling matrix() with a NumPy array will convert the array to a matrix. How do you know if you have a sparse matrix? Use Matplotlib's spy() method. We Create a 2-D Array in Numpy and call it a Matrix. When objects are deleted or go our of scope, the memory used for these variables isn't freed up until a garbage collection is performed. SciPy Tutorial. If is None, then the ordering is produced by G. 6k points) python. they are n-dimensional. dtype (NumPy data-type, optional) - A valid NumPy dtype used to initialize the array. If you rely on the fact that there is at most a constant number of non-zero elements in a single column, it is your responsibility to ensure that the matrix is symmetric. Background information ¶ Because in heterogeneous computing systems copying data across the bus from host memory to device memory (or vice versa) commonly incurs a proportionally substantial wait, PyViennaCL. Compressed Sparse Array¶ The csarray class represents a Compressed Sparse Array object which is essentially a 2d matrix or 1d vector with few nonzero elements. CSR and CSC are difficult to construct from scratch, while COO and DOK are easier to construct. True by default. Like and share. In this article, we are going to learn how to implement a sparse matrix for 3-tuple method using an array in the data structure? Submitted by Manu Jemini, on December 19, 2017 A sparse matrix is a matrix in which most of the elements are zero. array + scipy. If file is a string or Path, a. pyomo-latest/index. 行列を表すために,numpyではarrayとmatrixを使うことができる. しかし,掛け算の挙動などが,これら2つで異なるためにさらにややこしい印象がある. 自分用備忘録のためにarray,matrixそれぞれで特定の演算をするためにはどうすればいいかをまとめる.. matrix (which will be deprecated in NumPy at some point). they are n-dimensional. You're technically trying to index an uninitialized array. But the matrix is in 2d. csr_matrix that represents words in a document and a list of lists where each index represents the categories for each index in the matrix. SciPy's csc_matrix with a single column; We recommend using NumPy arrays over lists for efficiency, and using the factory methods implemented in Vectors to create sparse vectors. There are 7 different types of sparse matrices available. mmwrite / io. Download files. A different implementation would have to be written for. py file import tensorflow as tf import numpy as np We're going to begin by generating a NumPy array by using the random. In the example below, we define a 3 x 6 sparse matrix as a dense array, convert it to a CSR sparse representation, and then convert it back to a dense array by calling the todense() function. When feeding a numpy array versus a sparse matrix into the Scikit logistic regression classifier, it did not seem to make much of a difference, however. roll (a, shift[, axis]) Shifts elements of an array along specified axis. array + scipy. While this is the mathematical definition, I will be using the term sparse for matrices with only NNZ elements and dense for matrices with all elements. Creating a sparse Document Term Matrix for Topic Modeling via LDA June 17, 2016 11:23 am , Markus Konrad To do topic modeling with methods like Latent Dirichlet Allocation , it is necessary to build a Document Term Matrix (DTM) that contains the number of term occurrences per document. Backend for xESMF. Sparse Matrices in Python. A ((n, n) array or sparse matrix) - B ((n, p) array or sparse matrix) - build_inverse (bool, optional) - Whether to construct the inverse of the block matrix, as opposed to just. Using Numpy : Multiplication using Numpy also know as vectorization which main aim to reduce or remove the explicit use of for loops in the program by which computation becomes faster. Using the function above, I generated a benchmark for problem sizes ranging from 10 to 2,000, averaging computation times over 10 runs for each point. Given an index and pointer array, it is possible to tell beginning and end of each document. sort boolean, optional. find()¶ Returns three Numpy arrays to describe the sparsity pattern of self in so-called coordinate (or triplet) format:. True by default. NumPy 2D array(s), pandas DataFrame, H2O DataTable's Frame, SciPy sparse matrix; LightGBM binary file; The data is stored in a Dataset object. Let's say I also have a collection of scipy sparse matrices with the same dimensions as the numpy matrix. I'm using the SciPy sparse. , [1, 2, 3] and the following as sparse vectors: MLlib’s SparseVector. python arrays matlab scipy sparse-matrix |. For the sake of example, let's assume my symmetric sparse matrix is a 5x5: A = 1 -1 0 -3 0 -1 5 0 0 0 0 0 4 6 4 -3 0 6 7 0 0 0 4 0. If None, then the NumPy default is used. If file is a string or Path, a. CuPy is an open-source matrix library accelerated with NVIDIA CUDA. matrix which caused downstream problems. You can use numpy dense matrices. If the ratio of Number of Non-Zero elements to the size is less than 0. We'll also make use of the coo_matrix class from scipy. array(<50x5 sparse matrix of type '' with 50 stored elements in Compressed Sparse Column format>, dtype=object) I'm just a newbie who thought to use the usual pattern. It contains 2 rows and 3 columns. This builds on top of the scipy. csc_matrix, scipy. Parameters. By contrast, if most of the elements are nonzero, then the matrix is considered dense. xarray should only be used in higher-level APIs which interface with this low-level backend. So, in places below where you see “sparse matrix”, know that we really mean a “2D array” but, unlike a matrix, the array can be generalized to higher dimensions. In this article, we are going to learn how to implement a sparse matrix for 3-tuple method using an array in the data structure? Submitted by Manu Jemini, on December 19, 2017 A sparse matrix is a matrix in which most of the elements are zero. I ran into this problem a few months back. However, to future-proof ourselves from a soon-to-be deprecated 2D numpy matrix format, we’ll be leveraging the PyData Sparse package for all of our sparse nd-array needs. Note that the output is a numpy array, not a dataframe. NumPy is, just like SciPy, Scikit-Learn, Pandas, etc. convert_to_tensor(arg, dtype=tf. If ``is_sparse=False`` then P is a numpy array with a shape of ``(A, S, S)`` and R is a numpy array with a shape of ``(S, A)``. This NumPy stack has similar users to other applications such as MATLAB , GNU Octave , and Scilab. As illustrated below, the COO format may also be used to efficiently construct matrices. They come in a number of flavours. A dense matrix stored in a NumPy array can be converted into a sparse matrix using the CSR representation by calling the csr_matrix() function. Note: This is not binary compatible with scipy’s save_npz(). The Compressed Sparse Row matrix is vice versa. Each sample (i. Arrays The central feature of NumPy is the array object class. Compressed Sparse Array¶ The csarray class represents a Compressed Sparse Array object which is essentially a 2d matrix or 1d vector with few nonzero elements. In our solution, we created a NumPy array with two nonzero values, then converted it into a sparse matrix. This implements sparse arrays of arbitrary dimension on top of numpy and scipy. Passing these arrays will give CSR matrix and this is how csr_matrix function in scipy works. warning for NumPy users:. For example, if the dtypes are float16 and float32, the results dtype will be float32. Lets call this matrix Asp. matrix (which will be deprecated in NumPy at some point). Creating a Sparse Matrix. indices is array of column indices; data is array of corresponding nonzero values; indptr points to row starts in indices and data; length is n_row + 1, last item = number of values = length of both indices and data. Save a sparse matrix to disk in numpy’s. Well, if this were an ordinary Numpy array then we could write: np. sparse import * from scipy import *. 5, the matrix is sparse. sparse matrix? <. Most of the linear algebra Numpy and Scipy functions operate on Numpy and can also transparently operate on the Scipy sparse arrays. When I pass a MATLAB double array to a Java method that accepts double[] argument. We will then learn the basics of the numpy library to manipulate numpy arrays both as arrays and as matrices. T) Out[282]: <1000x1000 sparse matrix of type '' with 999964 stored elements in Compressed Sparse Row format>. Multiplication of two matrices X and Y is defined only if the number of columns in X is equal to the number of rows Y. sparse matrix constructors as the dtype argument. CSR and CSC are difficult to construct from scratch, while COO and DOK are easier to construct. NumPy and SciPy are two powerful Python packages, however, that enable the language to be used efficiently for scientific purposes. What we want is sparse arrays that act like numpy. base module hyperlearn. csr_matrix that represents words in a document and a list of lists where each index represents the categories for each index in the matrix. For most sparse types, out is required to be memory contiguous (either C or Fortran ordered). The input is a 1M x 100K random sparse matrix with density 0. NumPy 2D array. Here is an example: > > The lil_matrix is meant for supporting fancy indexing, but it is > not efficient for matrices operations such as inversion or > multiplication; you should transform your matrix to another format for > performing such operations. The more important attributes of an ndarray object are:. Sparse Graphs in Python Playing with Word Ladders. If the NumPy array has a single data type for each array entry it will be converted to an appropriate Python data type. The problem that I am having is that I need to randomly select N amount of rows from the data. A local matrix has integer-typed row and column indices and double-typed values, stored on a single machine. If specified, uses this array as the output buffer instead of allocating a new array to return. sparse matrix,. A special SparseIndex object tracks where data has been "sparsified". Mathematically this corresponds to pre-multiplying the matrix by the permutation matrix P and post-multiplying it by P^-1 = P^T, but this is not a computationally reasonable solution. The more general option that will allow you to reshape it an array to any shape that has the same number of elements as the original array is numpy. Executive summary¶. From your explanation, it sounds like you might have succeeded in writing out a valid file, but you just need to symbolize it in QGIS. Show first n rows. This will make much more sense in an example. Returns arr ndarray, 2-dimensional. array + scipy. The issue I am facing is with converting from sparse(i,j,v,m,n) to csr_matrix((data, (row_ind, col_ind)), [shape=(M, N)]). NPY_DOUBLE), and data is the pointer to the memory that has been previously allocated. matrix, so if scipy. Access to a sparse matrix created in MEX from MATLAB. sparse行列(疎行列)の計算(四則演算、逆行列、固有値など)や各種処理(連結や保存など)を行う方法について、以下の内容を説明する。疎行列の四則演算、行列積疎行列(scipy. Using the example of @hjpaul we get the following comparison code. NumPy 2D array(s), pandas DataFrame, H2O DataTable's Frame, SciPy sparse matrix; LightGBM binary file; The data is stored in a Dataset object. R matrices and arrays are converted automatically to and from NumPy arrays. matrix or array to scipy sparse matrix. We're imagining ourselves saying something like c = csr_matrix(array), where array is a dense numpy array. If the matrix is scipy. sparse matrices since it would make them non-sparse and would potentially crash the program with memory exhaustion problems. array, which only handles one-dimensional arrays and offers less functionality. Any ideas or insights that could solve the problem efficiently would be great as the sparse matrix product takes 2/3 of the time of the non-sparse. This was just an introduction into numpy matrices on how to get started and do basic manipulations. in sparse matrix to the identity without changing sparsity hand side NumPy array. Pandas data frame, and. Creating a Pandas DataFrame from a Numpy array: How do I specify the index column and column headers? asked Jul 27, 2019 in Data Science by sourav ( 17. csr_matrix(A) would construct a csr type matrix from a dense numpy array A, while sp. Python For Data Science Cheat Sheet SciPy - Linear Algebra Learn More Python for Data Science Interactively at www. All diagonals are stored using two arrays, one for data and one for diagonal offsets. Notes Sparse matrices can be used in arithmetic operations: they support addition, subtraction, multiplication, division, and matrix power. NPY_DOUBLE), and data is the pointer to the memory that has been previously allocated. Given data with very few non zero values you want. Using nested lists as a matrix works for simple computational tasks, however, there is a better way of working with matrices in Python using NumPy package. similarity_matrix ({scipy. As I said in the comments, the problem appears in multiply, which should produce a sparse matrix for sparse+dense inputs but doesn't. Python Forums on Bytes. Sparse Matrices in Python. scipy documentation: Convert a sparse matrix to a dense matrix using SciPy. But that behavior definitely looks most bizarre and counter-intuitive to me. A ((n, n) array or sparse matrix) - B ((n, p) array or sparse matrix) - build_inverse (bool, optional) - Whether to construct the inverse of the block matrix, as opposed to just. Syntax : numpy. float64'' with 2 stored elements in Compressed Sparse Row format This is because, for an arbitrary function, its application to a sparse matrix is not necessarily sparse. Calling matrix() with a NumPy array will convert the array to a matrix. T) Out[282]: <1000x1000 sparse matrix of type '' with 999964 stored elements in Compressed Sparse Row format>. 严格意义上讲ndarray数据类型应属数组而非矩阵,而matrix才是矩阵,这个在NumPy创建matrix一章里有讲述,是最基本的矩阵matrix创建方法,忘记了可以回头看看。. The row, col, and data elements are stored as numpy arrays. Note: This is not binary compatible with scipy’s save_npz(). Python For Data Science Cheat Sheet SciPy - Linear Algebra Learn More Python for Data Science Interactively at www. SciPy sparse matricies don’t support the same API as the NumPy ndarray, so most methods won’t work on the result. Whether transform should produce scipy. Additional outside tutorials exist, such as the Scipy Lecture Notes or Elegant SciPy. In the example below, we define a 3 x 6 sparse matrix as a dense array, convert it to a CSR sparse representation, and then convert it back to a dense array by calling the todense() function. Sparse Graphs in Python Playing with Word Ladders. The following are code examples for showing how to use scipy. Ironically the multiplication using numpy is faster. For the sake of example, let's assume my symmetric sparse matrix is a 5x5: A = 1 -1 0 -3 0 -1 5 0 0 0 0 0 4 6 4 -3 0 6 7 0 0 0 4 0. sort boolean, optional. sparse matrix? I''d like something that works li, ID #3867558. You can vote up the examples you like or vote down the ones you don't like. The sparse=False argument outputs a non-sparse matrix. In scipy, the implementation is not limited to main diagonal only. com SciPy DataCamp Learn Python for Data Science Interactively. If you are new to Python, you may be confused by some of the pythonic ways of accessing data, such as negative indexing and array slicing. I figured that I'd use scipy's 'sparse' package to reduce the storage overhead, but I'm a little confused about how to create arrays, not matrices. In this code, i, j and row_in, col_ind will be passed with an index array - idx of size(124416, 1), while v and data will be passed with a 2D array - D22 of size(290, 434) Matlab:. This NumPy stack has similar users to other applications such as MATLAB , GNU Octave , and Scilab. A dense matrix stored in a NumPy array can be converted into a sparse matrix using the CSR representation by calling the csr_matrix() function. This was just an introduction into numpy matrices on how to get started and do basic manipulations. There are many applications in which we deal with matrices that are mostly zeros. NumPy / SciPy / Pandas Cheat Sheet Select column. Pointer helps in understanding index and value arrays. As far as I can tell, there is no way to do this efficiently through python. True by default. This will make much more sense in an example. Constrained linear least squares in Python using scipy and cvxopt. save_npz (filename, matrix[, compressed]) Save a sparse matrix to disk in numpy's. For SciPy sparse matrix, one can use todense() or toarray() to transform to NumPy matrix or array. R matrices and arrays are converted automatically to and from NumPy arrays. Reset index, putting old index in column named index. Other methods currently implemented include: min, max, mean, sum, std, var, trace, diag, tranpose, dot, pdot (parallel dot product with ndarrays). Here is an example: import numpy as np from scipy. sparse matrix? <. Sparse arrays and multicategories¶ There is an alternative to numpy. This may require copying data and coercing values, which may be expensive. Using the example of @hjpaul we get the following comparison code. sparse matrices since it would make them non-sparse and would potentially crash the program with memory exhaustion problems. lil_matrix, you just need to print out the representation because the lil_matrix is implemented as a _sequence of non-zero elements_ i. Scikit-Learn returns a SciPy sparse matrix for ndarrays passed to transform. I have ellipsed some code here to focus our discussion. sparse import random matrix = random(1000000, 100000, density=0. Any ideas about how this could best be accomplished? Thanks in advance. convert_to_tensor(arg, dtype=tf. from mlxtend. If the NumPy array has a single data type for each array entry it will be converted to an appropriate Python data type. ndarray and numpy. By default, the dtype of the returned array will be the common NumPy dtype of all types in the DataFrame. The problem that I am having is that I need to randomly select N amount of rows from the data. linalg module hyperlearn. 001, containing 100M non-zero values: from scipy. It's not too different approach for writing the matrix, but seems convenient. If you're a scientist who programs with Python, this practical guide not only teaches you the fundamental parts of SciPy and libraries related … - Selection from Elegant SciPy [Book]. CSR and CSC are difficult to construct from scratch, while COO and DOK are easier to construct. It implements a more versatile variant of the widely-used Compressed Column (or Row) Storage scheme. Another difference is that numpy matrices are strictly 2-dimensional, while numpy arrays can be of any dimension, i. NumPy 2D array. Skip to main content 搜尋此網誌 Ftdxyku. For 1-D arrays, it is the inner product of. create numpy arrays, slice arrays, merge arrays, basic types of numpy arrays, reading and writing arrays to file, reading and writing sparse matrices to svmlight format. we can easily represent a dense matrix as an array: Get unlimited access to the best stories on Medium — and support writers while you’re at it. The main motivation for using arrays in this manner is speed. We’re imagining ourselves saying something like c = csr_matrix(array), where array is a dense numpy array. Hi all, I've been working quite a lot with sparse vectors and sparse matrices (basically as feature vectors in the context of machine learning), and have Numpy-discussion. sparse returns a numpy. Many of the examples in this page use functionality from numpy. find()¶ Returns three Numpy arrays to describe the sparsity pattern of self in so-called coordinate (or triplet) format:. The type of feature values. A Sparse Matrix (SM) is a popular data structure that is used to stored two-dimension… How to Compute the mean of a distribution using Python and Numpy? In the last post, we have defined a function to compute the numerical integration in…. Otherwise you'll wind up with a huge file. When passed a Dask Array, OneHotEncoder. Replace values in numpy array based on condition. matrix; This issue also applies to subtraction. We will then learn the basics of the numpy library to manipulate numpy arrays both as arrays and as matrices. just what you need. You should ask in the numpy mailing list. multiply(sp. fit but apparently it doesn't accept this datatype. , using the toarray() method of the class) first before applying the method. The problem that I am having is that I need to randomly select N amount of rows from the data. In simple words, suppose you have a 2-D matrix with hundreds of elements, where only a few of them contain a non-zero value. The input is a 1M x 100K random sparse matrix with density 0. weight: string or None optional. com SciPy DataCamp Learn Python for Data Science Interactively. What are the functions to do the inverse? 对于SciPy稀疏矩阵,可以使用todensity()或toarray()将其转换为NumPy矩阵或数组。. It’s not a sparse matrix (so our code path skips the conditional on line 7) and it’s not a tuple (so it skips the conditional on line 10). Many of the examples in this page use functionality from numpy. Sparse Matrices in Python. These are not necessarily sparse in the typical "mostly 0". A can be a scipy sparse matrix or a numpy array. In scipy, we can construct a sparse matrix using scipy. Quite simple, I guessed. How to create a sparse matrix in Python. In scipy, the implementation is not limited to main diagonal only. A Sparse Matrix (SM) is a popular data structure that is used to stored two-dimension… How to Compute the mean of a distribution using Python and Numpy? In the last post, we have defined a function to compute the numerical integration in…. matmul(arg, arg) + arg # The following. csr_matrix(v)) >>> W + s <2x3 sparse matrix of type '' with 2. If the NumPy array has a user-specified compound data type the names of the data fields will be used as attribute keys in the resulting NetworkX graph. Select row by label. A sparse matrix is a matrix that has a value of 0 for most elements. Apply OneHotEncoder on DataFrame: # apply OneHotEncoder on categorical feature columns X_ohe = ohe. The issue I am facing is with converting from sparse(i,j,v,m,n) to csr_matrix((data, (row_ind, col_ind)), [shape=(M, N)]). Machine learning data is represented as arrays. How to transform numpy. If a matrix contains many zeros, converting the matrix to sparse storage saves memory. Python’s SciPy library has a lot of options for creating, storing, and operating with Sparse matrices. create numpy arrays, slice arrays, merge arrays, basic types of numpy arrays, reading and writing arrays to file, reading and writing sparse matrices to svmlight format. These are not necessarily sparse in the typical "mostly 0". ) and storage type (row or column major format). Generate a random sparse multidimensional array. Compressed Sparse Row matrix otherwise The one-hot encoded boolean array of the input transactions, where the columns represent the unique items found in the input array in alphabetic order. SciPy and NumPy are able to help us with this easily. Questions: In scipy, we can construct a sparse matrix using scipy. Arrays make operations with large amounts of numeric data very fast and are. Multiple Matrix Multiplication in numpy « James Hensman's Weblog […]. todense() print ' index of the row that is modified: {0}'. Let's say I also have a collection of scipy sparse matrices with the same dimensions as the numpy matrix. Syntax : numpy. We provide only a brief overview of this format on this page; a complete description is provided in the paper The Matrix Market Formats: Initial Design [Gziped PostScript, 51 Kbytes] [PostScript, 189 Kbytes]. array; numpy. This builds on top of the scipy. It reads data from one. three NumPy arrays: indices, indptr, data. For SciPy sparse matrix, one can use todense() or toarray() to transform to NumPy matrix or array. sparse matrix (use CSR format if you want to avoid the burden of a copy / conversion). If you're a scientist who programs with Python, this practical guide not only teaches you the fundamental parts of SciPy and libraries related … - Selection from Elegant SciPy [Book]. SciPy 2D sparse array. the problem having need randomly select n amount of rows data. CuPy is an open-source matrix library accelerated with NVIDIA CUDA. Matlabs lsqlin and lsqnonneg in Python with sparse matrices. G (graph) - The NetworkX graph used to construct the NumPy matrix. This implementation uses a randomized SVD implementation and can handle both scipy. , using the toarray() method of the class) first before applying the method. <196980x43 sparse matrix of type '' with 70875 stored elements in Compressed Sparse Row format> It's an integer overflow due to using the same integer type. Hi all, I've been working quite a lot with sparse vectors and sparse matrices (basically as feature vectors in the context of machine learning), and have Numpy-discussion. nodelist (list, optional) - The rows and columns are ordered according to the nodes in. bsr_matrix: Block Sparse Row matrix. As an example, here's some Python code that uses NumPy to generate a random, sparse matrix in $\mathbf{R}^{\text{10,000}\times \text{10,000}}$ with 20,000 non-zero entries between 0 and 1. I suspect the question comes down to when to use a SciPy sparse matrix over a NumPy matrix, because in practice for any small matrix or a matrix with very few zeros, a numpy matrix is preferable, because it allows almost all operations that a nump. Creating a sparse matrix¶. How to Represent and Transpose a Sparse Matrix in C++? December 3, 2016 No Comments c / c++ , data structure , math , programming languages , tutorial A Sparse Matrix (SM) is a popular data structure that is used to stored two-dimension Matrix when the total the empty/zero elements are the majority in the matrix. com/file/d/1tNiTUTrv9e. I am wondering if there is an existing data structure for sparse 3d matrix / array (tensor) in Python? p. SciPy #3 齊藤 淳 Jun Saito @dukecyto Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Access to a sparse matrix created in MEX from MATLAB. array is not the same as the Standard Python Library class array. One thing that we can do here is to use SciPy to help solve an equation. Returns arr ndarray, 2-dimensional. dtype (NumPy data-type, optional) - A valid NumPy dtype used to initialize the array. Download files. If you're not sure which to choose, learn more about installing packages. csr_matrix(A) would construct a csr type matrix from a dense numpy array A, while sp. If None, then the NumPy default is used. csr_matrix , it is going to be transposed. sparse matrix? <. ) and storage type (row or column major format). mmwrite / io.