ó ÍÝn[c@sqddlmZdZdZdefd„ƒYZdefd„ƒYZedkrmd d lZej ƒnd S( i(txranges s t_matrixcBs‚eZdZd„Zd„Zed„Zd„Zed„Z d„Z d„Z d„Z d „Z d „Zd „Zd „Zd „Zd„Zd„Zd„ZeZd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„Zd„ZeeeddƒZ d„Z!d„Z"ee!e"ddƒZ#d„Z$ee$ƒZ%d „Z&d!„Z'ee'ƒZ(d"„Z)e)Z*d#„Z+RS($sm Numerical matrix. Specify the dimensions or the data as a nested list. Elements default to zero. Use a flat list to create a column vector easily. By default, only mpf is used to store the data. You can specify another type using force_type=type. It's possible to specify None. Make sure force_type(force_type()) is fast. Creating matrices ----------------- Matrices in mpmath are implemented using dictionaries. Only non-zero values are stored, so it is cheap to represent sparse matrices. The most basic way to create one is to use the ``matrix`` class directly. You can create an empty matrix specifying the dimensions: >>> from mpmath import * >>> mp.dps = 15 >>> matrix(2) matrix( [['0.0', '0.0'], ['0.0', '0.0']]) >>> matrix(2, 3) matrix( [['0.0', '0.0', '0.0'], ['0.0', '0.0', '0.0']]) Calling ``matrix`` with one dimension will create a square matrix. To access the dimensions of a matrix, use the ``rows`` or ``cols`` keyword: >>> A = matrix(3, 2) >>> A matrix( [['0.0', '0.0'], ['0.0', '0.0'], ['0.0', '0.0']]) >>> A.rows 3 >>> A.cols 2 You can also change the dimension of an existing matrix. This will set the new elements to 0. If the new dimension is smaller than before, the concerning elements are discarded: >>> A.rows = 2 >>> A matrix( [['0.0', '0.0'], ['0.0', '0.0']]) Internally ``mpmathify`` is used every time an element is set. This is done using the syntax A[row,column], counting from 0: >>> A = matrix(2) >>> A[1,1] = 1 + 1j >>> A matrix( [['0.0', '0.0'], ['0.0', mpc(real='1.0', imag='1.0')]]) You can use the keyword ``force_type`` to change the function which is called on every new element: >>> matrix(2, 5, force_type=int) # doctest: +SKIP matrix( [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0]]) A more comfortable way to create a matrix lets you use nested lists: >>> matrix([[1, 2], [3, 4]]) matrix( [['1.0', '2.0'], ['3.0', '4.0']]) If you want to preserve the type of the elements you can use ``force_type=None``: >>> matrix([[1, 2.5], [1j, mpf(2)]], force_type=None) matrix( [['1.0', '2.5'], [mpc(real='0.0', imag='1.0'), '2.0']]) Convenient advanced functions are available for creating various standard matrices, see ``zeros``, ``ones``, ``diag``, ``eye``, ``randmatrix`` and ``hilbert``. Vectors ....... Vectors may also be represented by the ``matrix`` class (with rows = 1 or cols = 1). For vectors there are some things which make life easier. A column vector can be created using a flat list, a row vectors using an almost flat nested list:: >>> matrix([1, 2, 3]) matrix( [['1.0'], ['2.0'], ['3.0']]) >>> matrix([[1, 2, 3]]) matrix( [['1.0', '2.0', '3.0']]) Optionally vectors can be accessed like lists, using only a single index:: >>> x = matrix([1, 2, 3]) >>> x[1] mpf('2.0') >>> x[1,0] mpf('2.0') Other ..... Like you probably expected, matrices can be printed:: >>> print randmatrix(3) # doctest:+SKIP [ 0.782963853573023 0.802057689719883 0.427895717335467] [0.0541876859348597 0.708243266653103 0.615134039977379] [ 0.856151514955773 0.544759264818486 0.686210904770947] Use ``nstr`` or ``nprint`` to specify the number of digits to print:: >>> nprint(randmatrix(5), 3) # doctest:+SKIP [2.07e-1 1.66e-1 5.06e-1 1.89e-1 8.29e-1] [6.62e-1 6.55e-1 4.47e-1 4.82e-1 2.06e-2] [4.33e-1 7.75e-1 6.93e-2 2.86e-1 5.71e-1] [1.01e-1 2.53e-1 6.13e-1 3.32e-1 2.59e-1] [1.56e-1 7.27e-2 6.05e-1 6.67e-2 2.79e-1] As matrices are mutable, you will need to copy them sometimes:: >>> A = matrix(2) >>> A matrix( [['0.0', '0.0'], ['0.0', '0.0']]) >>> B = A.copy() >>> B[0,0] = 1 >>> B matrix( [['1.0', '0.0'], ['0.0', '0.0']]) >>> A matrix( [['0.0', '0.0'], ['0.0', '0.0']]) Finally, it is possible to convert a matrix to a nested list. This is very useful, as most Python libraries involving matrices or arrays (namely NumPy or SymPy) support this format:: >>> B.tolist() [[mpf('1.0'), mpf('0.0')], [mpf('0.0'), mpf('0.0')]] Matrix operations ----------------- You can add and subtract matrices of compatible dimensions:: >>> A = matrix([[1, 2], [3, 4]]) >>> B = matrix([[-2, 4], [5, 9]]) >>> A + B matrix( [['-1.0', '6.0'], ['8.0', '13.0']]) >>> A - B matrix( [['3.0', '-2.0'], ['-2.0', '-5.0']]) >>> A + ones(3) # doctest:+ELLIPSIS Traceback (most recent call last): ... ValueError: incompatible dimensions for addition It is possible to multiply or add matrices and scalars. In the latter case the operation will be done element-wise:: >>> A * 2 matrix( [['2.0', '4.0'], ['6.0', '8.0']]) >>> A / 4 matrix( [['0.25', '0.5'], ['0.75', '1.0']]) >>> A - 1 matrix( [['0.0', '1.0'], ['2.0', '3.0']]) Of course you can perform matrix multiplication, if the dimensions are compatible:: >>> A * B matrix( [['8.0', '22.0'], ['14.0', '48.0']]) >>> matrix([[1, 2, 3]]) * matrix([[-6], [7], [-2]]) matrix( [['2.0']]) You can raise powers of square matrices:: >>> A**2 matrix( [['7.0', '10.0'], ['15.0', '22.0']]) Negative powers will calculate the inverse:: >>> A**-1 matrix( [['-2.0', '1.0'], ['1.5', '-0.5']]) >>> A * A**-1 matrix( [['1.0', '1.0842021724855e-19'], ['-2.16840434497101e-19', '1.0']]) Matrix transposition is straightforward:: >>> A = ones(2, 3) >>> A matrix( [['1.0', '1.0', '1.0'], ['1.0', '1.0', '1.0']]) >>> A.T matrix( [['1.0', '1.0'], ['1.0', '1.0'], ['1.0', '1.0']]) Norms ..... Sometimes you need to know how "large" a matrix or vector is. Due to their multidimensional nature it's not possible to compare them, but there are several functions to map a matrix or a vector to a positive real number, the so called norms. For vectors the p-norm is intended, usually the 1-, the 2- and the oo-norm are used. >>> x = matrix([-10, 2, 100]) >>> norm(x, 1) mpf('112.0') >>> norm(x, 2) mpf('100.5186549850325') >>> norm(x, inf) mpf('100.0') Please note that the 2-norm is the most used one, though it is more expensive to calculate than the 1- or oo-norm. It is possible to generalize some vector norms to matrix norm:: >>> A = matrix([[1, -1000], [100, 50]]) >>> mnorm(A, 1) mpf('1050.0') >>> mnorm(A, inf) mpf('1001.0') >>> mnorm(A, 'F') mpf('1006.2310867787777') The last norm (the "Frobenius-norm") is an approximation for the 2-norm, which is hard to calculate and not available. The Frobenius-norm lacks some mathematical properties you might expect from a norm. c OsÂi|_d|_|jd|jjƒ}|s<d„}nt|dttfƒrCt|ddttfƒrñ|d}t |ƒ|_ t |dƒ|_ xŸt |ƒD]?\}}x0t |ƒD]"\}}||ƒ|||f&siis expected intttolists#could not interpret given arguments(t _matrix__datatNonet_LUtgettctxtconvertt isinstancetlistttupletlent _matrix__rowst _matrix__colst enumeratetintt TypeErrorRtcopyRthasattrtmatrixR( tselftargstkwargsR tAtitrowtjtatvte((s7lib/python2.7/site-packages/mpmath/matrices/matrices.pyt__init__sN    !      +  cCss|jj|j|jƒ}xQt|jƒD]@}x7t|jƒD]&}||||fƒ|||f‰x5tˆjƒD]$‰ˆˆˆˆf|ˆˆfHs(R R RRRRURtfdot(RRZR$((RRRZRs7lib/python2.7/site-packages/mpmath/matrices/matrices.pyt__mul__@s(&cCs1t||jjƒr$tdƒ‚n|j|ƒS(Ns&other should not be type of ctx.matrix(R R RRR\(RRZ((s7lib/python2.7/site-packages/mpmath/matrices/matrices.pyt__rmul__SscCst|tƒstdƒ‚n|j|jks?tdƒ‚n|}|dkrd|jj|jƒS|dkr€| }t}nt}|}d}|j ƒ}xA|dkrá|ddkrÊ||}n||}|d}q¡W|rý|jj |ƒ}n|S(Ns$only integer exponents are supporteds*only powers of square matrices are definediii( R RRURRR teyeRCtFalseRtinverse(RRZR3tnegRRStz((s7lib/python2.7/site-packages/mpmath/matrices/matrices.pyt__pow__Ys,      cCst||jjƒ st‚|jj|j|jƒ}xOt|jƒD]>}x5t|jƒD]$}|||f||||ft||jjƒrÃ|j|jko6|j|jksHtdƒ‚n|jj|j|jƒ}xYt|jƒD]H}x?t|jƒD].}|||f|||f|||f>> from mpmath import diag, mp >>> mp.pretty = False >>> diag([1, 2, 3]) matrix( [['1.0', '0.0', '0.0'], ['0.0', '2.0', '0.0'], ['0.0', '0.0', '3.0']]) (RRR(R tdiagonalRRR((s7lib/python2.7/site-packages/mpmath/matrices/matrices.pytdiagðs cOs¹t|ƒdkr#|d}}n?t|ƒdkrL|d}|d}ntdt|ƒƒ‚|j|||}x;t|ƒD]-}x$t|ƒD]}d|||f>> from mpmath import zeros, mp >>> mp.pretty = False >>> zeros(2) matrix( [['0.0', '0.0'], ['0.0', '0.0']]) iiis*zeros expected at most 2 arguments, got %i(RRRR(R RRRRR3RRR((s7lib/python2.7/site-packages/mpmath/matrices/matrices.pytzeross   cOs¹t|ƒdkr#|d}}n?t|ƒdkrL|d}|d}ntdt|ƒƒ‚|j|||}x;t|ƒD]-}x$t|ƒD]}d|||f>> from mpmath import ones, mp >>> mp.pretty = False >>> ones(2) matrix( [['1.0', '1.0'], ['1.0', '1.0']]) iiis)ones expected at most 2 arguments, got %i(RRRR(R RRRRR3RRR((s7lib/python2.7/site-packages/mpmath/matrices/matrices.pytoness   cCsx|dkr|}n|j||ƒ}xJt|ƒD]<}x3t|ƒD]%}|j||d|||f= min and >> from mpmath import randmatrix >>> randmatrix(2) # doctest:+SKIP matrix( [['0.53491598236191806', '0.57195669543302752'], ['0.85589992269513615', '0.82444367501382143']]) (RRtrand( R RRR3tminR.RRRR((s7lib/python2.7/site-packages/mpmath/matrices/matrices.pyt randmatrixFs *cCs³||krdSt||jƒrtxŠt|jƒD];}|||f|||f|||f<|||f>> from mpmath import * >>> mp.dps = 15; mp.pretty = False >>> x = matrix([-10, 2, 100]) >>> norm(x, 1) mpf('112.0') >>> norm(x, 2) mpf('100.5186549850325') >>> norm(x, inf) mpf('100.0') c3s|]}ˆj|ƒVqdS(N(tabsmax(RXR(R (s7lib/python2.7/site-packages/mpmath/matrices/matrices.pys ¡sitabsoluteitsquaredc3s|]}t|ƒˆVqdS(N(tabs(RXR(tp(s7lib/python2.7/site-packages/mpmath/matrices/matrices.pys §ssp has to be >= 1N( titerRR‘R„RR tinfR.tfsumtsqrttnthrootRU(R RR•((R R•s7lib/python2.7/site-packages/mpmath/matrices/matrices.pytnormxs"   " )csôˆjˆƒ‰t|ƒtk rjt|ƒtkrXdj|jƒƒrXˆjˆdƒSˆj|ƒ}nˆjˆj ‰‰|dkr¯t ‡‡‡fd†t ˆƒDƒƒS|ˆj krät ‡‡‡fd†t ˆƒDƒƒSt dƒ‚dS(só Gives the matrix (operator) `p`-norm of A. Currently ``p=1`` and ``p=inf`` are supported: ``p=1`` gives the 1-norm (maximal column sum) ``p=inf`` gives the `\infty`-norm (maximal row sum). You can use the string 'inf' as well as float('inf') or mpf('inf') ``p=2`` (not implemented) for a square matrix is the usual spectral matrix norm, i.e. the largest singular value. ``p='f'`` (or 'F', 'fro', 'Frobenius, 'frobenius') gives the Frobenius norm, which is the elementwise 2-norm. The Frobenius norm is an approximation of the spectral norm and satisfies .. math :: \frac{1}{\sqrt{\mathrm{rank}(A)}} \|A\|_F \le \|A\|_2 \le \|A\|_F The Frobenius norm lacks some mathematical properties that might be expected of a norm. For general elementwise `p`-norms, use :func:`~mpmath.norm` instead. **Examples** >>> from mpmath import * >>> mp.dps = 15; mp.pretty = False >>> A = matrix([[1, -1000], [100, 50]]) >>> mnorm(A, 1) mpf('1050.0') >>> mnorm(A, inf) mpf('1001.0') >>> mnorm(A, 'F') mpf('1006.2310867787777') t frobeniusiic3s=|]3‰ˆj‡‡fd†tˆƒDƒddƒVqdS(c3s|]}ˆ|ˆfVqdS(N((RXR(RR(s7lib/python2.7/site-packages/mpmath/matrices/matrices.pys ÙsR’iN(R˜R(RX(RR RR(Rs7lib/python2.7/site-packages/mpmath/matrices/matrices.pys Ùsc3s=|]3‰ˆj‡‡fd†tˆƒDƒddƒVqdS(c3s|]}ˆˆ|fVqdS(N((RXR(RR(s7lib/python2.7/site-packages/mpmath/matrices/matrices.pys ÛsR’iN(R˜R(RX(RR R3(Rs7lib/python2.7/site-packages/mpmath/matrices/matrices.pys Ûssmatrix p-norm for arbitrary pN(RR„RR-t startswithtlowerR›R R*R(R.RR—tNotImplementedError(R RR•((RR RRR3s7lib/python2.7/site-packages/mpmath/matrices/matrices.pytmnorm«s'' &&N(RwRxR"R^R†R‡RˆRRŠRRŽRR›R (((s7lib/python2.7/site-packages/mpmath/matrices/matrices.pyRƒßs        3t__main__iÿÿÿÿN( t libmp.backendRR2R0tobjectRRƒRwtdoctestttestmod(((s7lib/python2.7/site-packages/mpmath/matrices/matrices.pytsÿÿÙÿ