Source

numpypy / numpypy / linalg / linalg.py

Full commit
   1
   2
   3
   4
   5
   6
   7
   8
   9
  10
  11
  12
  13
  14
  15
  16
  17
  18
  19
  20
  21
  22
  23
  24
  25
  26
  27
  28
  29
  30
  31
  32
  33
  34
  35
  36
  37
  38
  39
  40
  41
  42
  43
  44
  45
  46
  47
  48
  49
  50
  51
  52
  53
  54
  55
  56
  57
  58
  59
  60
  61
  62
  63
  64
  65
  66
  67
  68
  69
  70
  71
  72
  73
  74
  75
  76
  77
  78
  79
  80
  81
  82
  83
  84
  85
  86
  87
  88
  89
  90
  91
  92
  93
  94
  95
  96
  97
  98
  99
 100
 101
 102
 103
 104
 105
 106
 107
 108
 109
 110
 111
 112
 113
 114
 115
 116
 117
 118
 119
 120
 121
 122
 123
 124
 125
 126
 127
 128
 129
 130
 131
 132
 133
 134
 135
 136
 137
 138
 139
 140
 141
 142
 143
 144
 145
 146
 147
 148
 149
 150
 151
 152
 153
 154
 155
 156
 157
 158
 159
 160
 161
 162
 163
 164
 165
 166
 167
 168
 169
 170
 171
 172
 173
 174
 175
 176
 177
 178
 179
 180
 181
 182
 183
 184
 185
 186
 187
 188
 189
 190
 191
 192
 193
 194
 195
 196
 197
 198
 199
 200
 201
 202
 203
 204
 205
 206
 207
 208
 209
 210
 211
 212
 213
 214
 215
 216
 217
 218
 219
 220
 221
 222
 223
 224
 225
 226
 227
 228
 229
 230
 231
 232
 233
 234
 235
 236
 237
 238
 239
 240
 241
 242
 243
 244
 245
 246
 247
 248
 249
 250
 251
 252
 253
 254
 255
 256
 257
 258
 259
 260
 261
 262
 263
 264
 265
 266
 267
 268
 269
 270
 271
 272
 273
 274
 275
 276
 277
 278
 279
 280
 281
 282
 283
 284
 285
 286
 287
 288
 289
 290
 291
 292
 293
 294
 295
 296
 297
 298
 299
 300
 301
 302
 303
 304
 305
 306
 307
 308
 309
 310
 311
 312
 313
 314
 315
 316
 317
 318
 319
 320
 321
 322
 323
 324
 325
 326
 327
 328
 329
 330
 331
 332
 333
 334
 335
 336
 337
 338
 339
 340
 341
 342
 343
 344
 345
 346
 347
 348
 349
 350
 351
 352
 353
 354
 355
 356
 357
 358
 359
 360
 361
 362
 363
 364
 365
 366
 367
 368
 369
 370
 371
 372
 373
 374
 375
 376
 377
 378
 379
 380
 381
 382
 383
 384
 385
 386
 387
 388
 389
 390
 391
 392
 393
 394
 395
 396
 397
 398
 399
 400
 401
 402
 403
 404
 405
 406
 407
 408
 409
 410
 411
 412
 413
 414
 415
 416
 417
 418
 419
 420
 421
 422
 423
 424
 425
 426
 427
 428
 429
 430
 431
 432
 433
 434
 435
 436
 437
 438
 439
 440
 441
 442
 443
 444
 445
 446
 447
 448
 449
 450
 451
 452
 453
 454
 455
 456
 457
 458
 459
 460
 461
 462
 463
 464
 465
 466
 467
 468
 469
 470
 471
 472
 473
 474
 475
 476
 477
 478
 479
 480
 481
 482
 483
 484
 485
 486
 487
 488
 489
 490
 491
 492
 493
 494
 495
 496
 497
 498
 499
 500
 501
 502
 503
 504
 505
 506
 507
 508
 509
 510
 511
 512
 513
 514
 515
 516
 517
 518
 519
 520
 521
 522
 523
 524
 525
 526
 527
 528
 529
 530
 531
 532
 533
 534
 535
 536
 537
 538
 539
 540
 541
 542
 543
 544
 545
 546
 547
 548
 549
 550
 551
 552
 553
 554
 555
 556
 557
 558
 559
 560
 561
 562
 563
 564
 565
 566
 567
 568
 569
 570
 571
 572
 573
 574
 575
 576
 577
 578
 579
 580
 581
 582
 583
 584
 585
 586
 587
 588
 589
 590
 591
 592
 593
 594
 595
 596
 597
 598
 599
 600
 601
 602
 603
 604
 605
 606
 607
 608
 609
 610
 611
 612
 613
 614
 615
 616
 617
 618
 619
 620
 621
 622
 623
 624
 625
 626
 627
 628
 629
 630
 631
 632
 633
 634
 635
 636
 637
 638
 639
 640
 641
 642
 643
 644
 645
 646
 647
 648
 649
 650
 651
 652
 653
 654
 655
 656
 657
 658
 659
 660
 661
 662
 663
 664
 665
 666
 667
 668
 669
 670
 671
 672
 673
 674
 675
 676
 677
 678
 679
 680
 681
 682
 683
 684
 685
 686
 687
 688
 689
 690
 691
 692
 693
 694
 695
 696
 697
 698
 699
 700
 701
 702
 703
 704
 705
 706
 707
 708
 709
 710
 711
 712
 713
 714
 715
 716
 717
 718
 719
 720
 721
 722
 723
 724
 725
 726
 727
 728
 729
 730
 731
 732
 733
 734
 735
 736
 737
 738
 739
 740
 741
 742
 743
 744
 745
 746
 747
 748
 749
 750
 751
 752
 753
 754
 755
 756
 757
 758
 759
 760
 761
 762
 763
 764
 765
 766
 767
 768
 769
 770
 771
 772
 773
 774
 775
 776
 777
 778
 779
 780
 781
 782
 783
 784
 785
 786
 787
 788
 789
 790
 791
 792
 793
 794
 795
 796
 797
 798
 799
 800
 801
 802
 803
 804
 805
 806
 807
 808
 809
 810
 811
 812
 813
 814
 815
 816
 817
 818
 819
 820
 821
 822
 823
 824
 825
 826
 827
 828
 829
 830
 831
 832
 833
 834
 835
 836
 837
 838
 839
 840
 841
 842
 843
 844
 845
 846
 847
 848
 849
 850
 851
 852
 853
 854
 855
 856
 857
 858
 859
 860
 861
 862
 863
 864
 865
 866
 867
 868
 869
 870
 871
 872
 873
 874
 875
 876
 877
 878
 879
 880
 881
 882
 883
 884
 885
 886
 887
 888
 889
 890
 891
 892
 893
 894
 895
 896
 897
 898
 899
 900
 901
 902
 903
 904
 905
 906
 907
 908
 909
 910
 911
 912
 913
 914
 915
 916
 917
 918
 919
 920
 921
 922
 923
 924
 925
 926
 927
 928
 929
 930
 931
 932
 933
 934
 935
 936
 937
 938
 939
 940
 941
 942
 943
 944
 945
 946
 947
 948
 949
 950
 951
 952
 953
 954
 955
 956
 957
 958
 959
 960
 961
 962
 963
 964
 965
 966
 967
 968
 969
 970
 971
 972
 973
 974
 975
 976
 977
 978
 979
 980
 981
 982
 983
 984
 985
 986
 987
 988
 989
 990
 991
 992
 993
 994
 995
 996
 997
 998
 999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
"""Lite version of scipy.linalg.

Notes
-----
This module is a lite version of the linalg.py module in SciPy which
contains high-level Python interface to the LAPACK library.  The lite
version only accesses the following LAPACK functions: dgesv, zgesv,
dgeev, zgeev, dgesdd, zgesdd, dgelsd, zgelsd, dsyevd, zheevd, dgetrf,
zgetrf, dpotrf, zpotrf, dgeqrf, zgeqrf, zungqr, dorgqr.
"""

__all__ = ['matrix_power', 'solve', 'tensorsolve', 'tensorinv', 'inv',
           'cholesky', 'eigvals', 'eigvalsh', 'pinv', 'slogdet', 'det',
           'svd', 'eig', 'eigh','lstsq', 'norm', 'qr', 'cond', 'matrix_rank',
           'LinAlgError']

from numpy.core import array, asarray, zeros, empty, transpose, \
        intc, single, double, csingle, cdouble, inexact, complexfloating, \
        newaxis, ravel, all, Inf, dot, add, multiply, identity, sqrt, \
        maximum, flatnonzero, diagonal, arange, fastCopyAndTranspose, sum, \
        isfinite, size, finfo, absolute, log, exp
from numpy.lib import triu
from numpy.linalg import lapack_lite
from numpy.matrixlib.defmatrix import matrix_power
from numpy.compat import asbytes

# For Python2/3 compatibility
_N = asbytes('N')
_V = asbytes('V')
_A = asbytes('A')
_S = asbytes('S')
_L = asbytes('L')

fortran_int = intc

# Error object
class LinAlgError(Exception):
    """
    Generic Python-exception-derived object raised by linalg functions.

    General purpose exception class, derived from Python's exception.Exception
    class, programmatically raised in linalg functions when a Linear
    Algebra-related condition would prevent further correct execution of the
    function.

    Parameters
    ----------
    None

    Examples
    --------
    >>> from numpy import linalg as LA
    >>> LA.inv(np.zeros((2,2)))
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
      File "...linalg.py", line 350,
        in inv return wrap(solve(a, identity(a.shape[0], dtype=a.dtype)))
      File "...linalg.py", line 249,
        in solve
        raise LinAlgError('Singular matrix')
    numpy.linalg.LinAlgError: Singular matrix

    """
    pass

def _makearray(a):
    new = asarray(a)
    wrap = getattr(a, "__array_prepare__", new.__array_wrap__)
    return new, wrap

def isComplexType(t):
    return issubclass(t, complexfloating)

_real_types_map = {single : single,
                   double : double,
                   csingle : single,
                   cdouble : double}

_complex_types_map = {single : csingle,
                      double : cdouble,
                      csingle : csingle,
                      cdouble : cdouble}

def _realType(t, default=double):
    return _real_types_map.get(t, default)

def _complexType(t, default=cdouble):
    return _complex_types_map.get(t, default)

def _linalgRealType(t):
    """Cast the type t to either double or cdouble."""
    return double

_complex_types_map = {single : csingle,
                      double : cdouble,
                      csingle : csingle,
                      cdouble : cdouble}

def _commonType(*arrays):
    # in lite version, use higher precision (always double or cdouble)
    result_type = single
    is_complex = False
    for a in arrays:
        if issubclass(a.dtype.type, inexact):
            if isComplexType(a.dtype.type):
                is_complex = True
            rt = _realType(a.dtype.type, default=None)
            if rt is None:
                # unsupported inexact scalar
                raise TypeError("array type %s is unsupported in linalg" %
                        (a.dtype.name,))
        else:
            rt = double
        if rt is double:
            result_type = double
    if is_complex:
        t = cdouble
        result_type = _complex_types_map[result_type]
    else:
        t = double
    return t, result_type

# _fastCopyAndTranpose assumes the input is 2D (as all the calls in here are).

_fastCT = fastCopyAndTranspose

def _to_native_byte_order(*arrays):
    ret = []
    for arr in arrays:
        if arr.dtype.byteorder not in ('=', '|'):
            ret.append(asarray(arr, dtype=arr.dtype.newbyteorder('=')))
        else:
            ret.append(arr)
    if len(ret) == 1:
        return ret[0]
    else:
        return ret

def _fastCopyAndTranspose(type, *arrays):
    cast_arrays = ()
    for a in arrays:
        if a.dtype.type is type:
            cast_arrays = cast_arrays + (_fastCT(a),)
        else:
            cast_arrays = cast_arrays + (_fastCT(a.astype(type)),)
    if len(cast_arrays) == 1:
        return cast_arrays[0]
    else:
        return cast_arrays

def _assertRank2(*arrays):
    for a in arrays:
        if len(a.shape) != 2:
            raise LinAlgError('%d-dimensional array given. Array must be '
                    'two-dimensional' % len(a.shape))

def _assertSquareness(*arrays):
    for a in arrays:
        if max(a.shape) != min(a.shape):
            raise LinAlgError('Array must be square')

def _assertFinite(*arrays):
    for a in arrays:
        if not (isfinite(a).all()):
            raise LinAlgError("Array must not contain infs or NaNs")

def _assertNonEmpty(*arrays):
    for a in arrays:
        if size(a) == 0:
            raise LinAlgError("Arrays cannot be empty")


# Linear equations

def tensorsolve(a, b, axes=None):
    """
    Solve the tensor equation ``a x = b`` for x.

    It is assumed that all indices of `x` are summed over in the product,
    together with the rightmost indices of `a`, as is done in, for example,
    ``tensordot(a, x, axes=len(b.shape))``.

    Parameters
    ----------
    a : array_like
        Coefficient tensor, of shape ``b.shape + Q``. `Q`, a tuple, equals
        the shape of that sub-tensor of `a` consisting of the appropriate
        number of its rightmost indices, and must be such that
       ``prod(Q) == prod(b.shape)`` (in which sense `a` is said to be
       'square').
    b : array_like
        Right-hand tensor, which can be of any shape.
    axes : tuple of ints, optional
        Axes in `a` to reorder to the right, before inversion.
        If None (default), no reordering is done.

    Returns
    -------
    x : ndarray, shape Q

    Raises
    ------
    LinAlgError
        If `a` is singular or not 'square' (in the above sense).

    See Also
    --------
    tensordot, tensorinv, einsum

    Examples
    --------
    >>> a = np.eye(2*3*4)
    >>> a.shape = (2*3, 4, 2, 3, 4)
    >>> b = np.random.randn(2*3, 4)
    >>> x = np.linalg.tensorsolve(a, b)
    >>> x.shape
    (2, 3, 4)
    >>> np.allclose(np.tensordot(a, x, axes=3), b)
    True

    """
    a,wrap = _makearray(a)
    b = asarray(b)
    an = a.ndim

    if axes is not None:
        allaxes = range(0, an)
        for k in axes:
            allaxes.remove(k)
            allaxes.insert(an, k)
        a = a.transpose(allaxes)

    oldshape = a.shape[-(an-b.ndim):]
    prod = 1
    for k in oldshape:
        prod *= k

    a = a.reshape(-1, prod)
    b = b.ravel()
    res = wrap(solve(a, b))
    res.shape = oldshape
    return res

def solve(a, b):
    """
    Solve a linear matrix equation, or system of linear scalar equations.

    Computes the "exact" solution, `x`, of the well-determined, i.e., full
    rank, linear matrix equation `ax = b`.

    Parameters
    ----------
    a : array_like, shape (M, M)
        Coefficient matrix.
    b : array_like, shape (M,) or (M, N)
        Ordinate or "dependent variable" values.

    Returns
    -------
    x : ndarray, shape (M,) or (M, N) depending on b
        Solution to the system a x = b

    Raises
    ------
    LinAlgError
        If `a` is singular or not square.

    Notes
    -----
    `solve` is a wrapper for the LAPACK routines `dgesv`_ and
    `zgesv`_, the former being used if `a` is real-valued, the latter if
    it is complex-valued.  The solution to the system of linear equations
    is computed using an LU decomposition [1]_ with partial pivoting and
    row interchanges.

    .. _dgesv: http://www.netlib.org/lapack/double/dgesv.f

    .. _zgesv: http://www.netlib.org/lapack/complex16/zgesv.f

    `a` must be square and of full-rank, i.e., all rows (or, equivalently,
    columns) must be linearly independent; if either is not true, use
    `lstsq` for the least-squares best "solution" of the
    system/equation.

    References
    ----------
    .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
           FL, Academic Press, Inc., 1980, pg. 22.

    Examples
    --------
    Solve the system of equations ``3 * x0 + x1 = 9`` and ``x0 + 2 * x1 = 8``:

    >>> a = np.array([[3,1], [1,2]])
    >>> b = np.array([9,8])
    >>> x = np.linalg.solve(a, b)
    >>> x
    array([ 2.,  3.])

    Check that the solution is correct:

    >>> (np.dot(a, x) == b).all()
    True

    """
    a, _ = _makearray(a)
    b, wrap = _makearray(b)
    one_eq = len(b.shape) == 1
    if one_eq:
        b = b[:, newaxis]
    _assertRank2(a, b)
    _assertSquareness(a)
    n_eq = a.shape[0]
    n_rhs = b.shape[1]
    if n_eq != b.shape[0]:
        raise LinAlgError('Incompatible dimensions')
    t, result_t = _commonType(a, b)
#    lapack_routine = _findLapackRoutine('gesv', t)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgesv
    else:
        lapack_routine = lapack_lite.dgesv
    a, b = _fastCopyAndTranspose(t, a, b)
    a, b = _to_native_byte_order(a, b)
    pivots = zeros(n_eq, fortran_int)
    results = lapack_routine(n_eq, n_rhs, a, n_eq, pivots, b, n_eq, 0)
    if results['info'] > 0:
        raise LinAlgError('Singular matrix')
    if one_eq:
        return wrap(b.ravel().astype(result_t))
    else:
        return wrap(b.transpose().astype(result_t))


def tensorinv(a, ind=2):
    """
    Compute the 'inverse' of an N-dimensional array.

    The result is an inverse for `a` relative to the tensordot operation
    ``tensordot(a, b, ind)``, i. e., up to floating-point accuracy,
    ``tensordot(tensorinv(a), a, ind)`` is the "identity" tensor for the
    tensordot operation.

    Parameters
    ----------
    a : array_like
        Tensor to 'invert'. Its shape must be 'square', i. e.,
        ``prod(a.shape[:ind]) == prod(a.shape[ind:])``.
    ind : int, optional
        Number of first indices that are involved in the inverse sum.
        Must be a positive integer, default is 2.

    Returns
    -------
    b : ndarray
        `a`'s tensordot inverse, shape ``a.shape[:ind] + a.shape[ind:]``.

    Raises
    ------
    LinAlgError
        If `a` is singular or not 'square' (in the above sense).

    See Also
    --------
    tensordot, tensorsolve

    Examples
    --------
    >>> a = np.eye(4*6)
    >>> a.shape = (4, 6, 8, 3)
    >>> ainv = np.linalg.tensorinv(a, ind=2)
    >>> ainv.shape
    (8, 3, 4, 6)
    >>> b = np.random.randn(4, 6)
    >>> np.allclose(np.tensordot(ainv, b), np.linalg.tensorsolve(a, b))
    True

    >>> a = np.eye(4*6)
    >>> a.shape = (24, 8, 3)
    >>> ainv = np.linalg.tensorinv(a, ind=1)
    >>> ainv.shape
    (8, 3, 24)
    >>> b = np.random.randn(24)
    >>> np.allclose(np.tensordot(ainv, b, 1), np.linalg.tensorsolve(a, b))
    True

    """
    a = asarray(a)
    oldshape = a.shape
    prod = 1
    if ind > 0:
        invshape = oldshape[ind:] + oldshape[:ind]
        for k in oldshape[ind:]:
            prod *= k
    else:
        raise ValueError("Invalid ind argument.")
    a = a.reshape(prod, -1)
    ia = inv(a)
    return ia.reshape(*invshape)


# Matrix inversion

def inv(a):
    """
    Compute the (multiplicative) inverse of a matrix.

    Given a square matrix `a`, return the matrix `ainv` satisfying
    ``dot(a, ainv) = dot(ainv, a) = eye(a.shape[0])``.

    Parameters
    ----------
    a : array_like, shape (M, M)
        Matrix to be inverted.

    Returns
    -------
    ainv : ndarray or matrix, shape (M, M)
        (Multiplicative) inverse of the matrix `a`.

    Raises
    ------
    LinAlgError
        If `a` is singular or not square.

    Examples
    --------
    >>> from numpy import linalg as LA
    >>> a = np.array([[1., 2.], [3., 4.]])
    >>> ainv = LA.inv(a)
    >>> np.allclose(np.dot(a, ainv), np.eye(2))
    True
    >>> np.allclose(np.dot(ainv, a), np.eye(2))
    True

    If a is a matrix object, then the return value is a matrix as well:

    >>> ainv = LA.inv(np.matrix(a))
    >>> ainv
    matrix([[-2. ,  1. ],
            [ 1.5, -0.5]])

    """
    a, wrap = _makearray(a)
    return wrap(solve(a, identity(a.shape[0], dtype=a.dtype)))


# Cholesky decomposition

def cholesky(a):
    """
    Cholesky decomposition.

    Return the Cholesky decomposition, `L * L.H`, of the square matrix `a`,
    where `L` is lower-triangular and .H is the conjugate transpose operator
    (which is the ordinary transpose if `a` is real-valued).  `a` must be
    Hermitian (symmetric if real-valued) and positive-definite.  Only `L` is
    actually returned.

    Parameters
    ----------
    a : array_like, shape (M, M)
        Hermitian (symmetric if all elements are real), positive-definite
        input matrix.

    Returns
    -------
    L : ndarray, or matrix object if `a` is, shape (M, M)
        Lower-triangular Cholesky factor of a.

    Raises
    ------
    LinAlgError
       If the decomposition fails, for example, if `a` is not
       positive-definite.

    Notes
    -----
    The Cholesky decomposition is often used as a fast way of solving

    .. math:: A \\mathbf{x} = \\mathbf{b}

    (when `A` is both Hermitian/symmetric and positive-definite).

    First, we solve for :math:`\\mathbf{y}` in

    .. math:: L \\mathbf{y} = \\mathbf{b},

    and then for :math:`\\mathbf{x}` in

    .. math:: L.H \\mathbf{x} = \\mathbf{y}.

    Examples
    --------
    >>> A = np.array([[1,-2j],[2j,5]])
    >>> A
    array([[ 1.+0.j,  0.-2.j],
           [ 0.+2.j,  5.+0.j]])
    >>> L = np.linalg.cholesky(A)
    >>> L
    array([[ 1.+0.j,  0.+0.j],
           [ 0.+2.j,  1.+0.j]])
    >>> np.dot(L, L.T.conj()) # verify that L * L.H = A
    array([[ 1.+0.j,  0.-2.j],
           [ 0.+2.j,  5.+0.j]])
    >>> A = [[1,-2j],[2j,5]] # what happens if A is only array_like?
    >>> np.linalg.cholesky(A) # an ndarray object is returned
    array([[ 1.+0.j,  0.+0.j],
           [ 0.+2.j,  1.+0.j]])
    >>> # But a matrix object is returned if A is a matrix object
    >>> LA.cholesky(np.matrix(A))
    matrix([[ 1.+0.j,  0.+0.j],
            [ 0.+2.j,  1.+0.j]])

    """
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    a = _fastCopyAndTranspose(t, a)
    a = _to_native_byte_order(a)
    m = a.shape[0]
    n = a.shape[1]
    if isComplexType(t):
        lapack_routine = lapack_lite.zpotrf
    else:
        lapack_routine = lapack_lite.dpotrf
    results = lapack_routine(_L, n, a, m, 0)
    if results['info'] > 0:
        raise LinAlgError('Matrix is not positive definite - '
                'Cholesky decomposition cannot be computed')
    s = triu(a, k=0).transpose()
    if (s.dtype != result_t):
        s = s.astype(result_t)
    return wrap(s)

# QR decompostion

def qr(a, mode='full'):
    """
    Compute the qr factorization of a matrix.

    Factor the matrix `a` as *qr*, where `q` is orthonormal and `r` is
    upper-triangular.

    Parameters
    ----------
    a : array_like
        Matrix to be factored, of shape (M, N).
    mode : {'full', 'r', 'economic'}, optional
        Specifies the values to be returned. 'full' is the default.
        Economic mode is slightly faster then 'r' mode if only `r` is needed.

    Returns
    -------
    q : ndarray of float or complex, optional
        The orthonormal matrix, of shape (M, K). Only returned if
        ``mode='full'``.
    r : ndarray of float or complex, optional
        The upper-triangular matrix, of shape (K, N) with K = min(M, N).
        Only returned when ``mode='full'`` or ``mode='r'``.
    a2 : ndarray of float or complex, optional
        Array of shape (M, N), only returned when ``mode='economic``'.
        The  diagonal and the upper triangle of `a2` contains `r`, while
        the rest of the matrix is undefined.

    Raises
    ------
    LinAlgError
        If factoring fails.

    Notes
    -----
    This is an interface to the LAPACK routines dgeqrf, zgeqrf,
    dorgqr, and zungqr.

    For more information on the qr factorization, see for example:
    http://en.wikipedia.org/wiki/QR_factorization

    Subclasses of `ndarray` are preserved, so if `a` is of type `matrix`,
    all the return values will be matrices too.

    Examples
    --------
    >>> a = np.random.randn(9, 6)
    >>> q, r = np.linalg.qr(a)
    >>> np.allclose(a, np.dot(q, r))  # a does equal qr
    True
    >>> r2 = np.linalg.qr(a, mode='r')
    >>> r3 = np.linalg.qr(a, mode='economic')
    >>> np.allclose(r, r2)  # mode='r' returns the same r as mode='full'
    True
    >>> # But only triu parts are guaranteed equal when mode='economic'
    >>> np.allclose(r, np.triu(r3[:6,:6], k=0))
    True

    Example illustrating a common use of `qr`: solving of least squares
    problems

    What are the least-squares-best `m` and `y0` in ``y = y0 + mx`` for
    the following data: {(0,1), (1,0), (1,2), (2,1)}. (Graph the points
    and you'll see that it should be y0 = 0, m = 1.)  The answer is provided
    by solving the over-determined matrix equation ``Ax = b``, where::

      A = array([[0, 1], [1, 1], [1, 1], [2, 1]])
      x = array([[y0], [m]])
      b = array([[1], [0], [2], [1]])

    If A = qr such that q is orthonormal (which is always possible via
    Gram-Schmidt), then ``x = inv(r) * (q.T) * b``.  (In numpy practice,
    however, we simply use `lstsq`.)

    >>> A = np.array([[0, 1], [1, 1], [1, 1], [2, 1]])
    >>> A
    array([[0, 1],
           [1, 1],
           [1, 1],
           [2, 1]])
    >>> b = np.array([1, 0, 2, 1])
    >>> q, r = LA.qr(A)
    >>> p = np.dot(q.T, b)
    >>> np.dot(LA.inv(r), p)
    array([  1.1e-16,   1.0e+00])

    """
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertNonEmpty(a)
    m, n = a.shape
    t, result_t = _commonType(a)
    a = _fastCopyAndTranspose(t, a)
    a = _to_native_byte_order(a)
    mn = min(m, n)
    tau = zeros((mn,), t)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgeqrf
        routine_name = 'zgeqrf'
    else:
        lapack_routine = lapack_lite.dgeqrf
        routine_name = 'dgeqrf'

    # calculate optimal size of work data 'work'
    lwork = 1
    work = zeros((lwork,), t)
    results = lapack_routine(m, n, a, m, tau, work, -1, 0)
    if results['info'] != 0:
        raise LinAlgError('%s returns %d' % (routine_name, results['info']))

    # do qr decomposition
    lwork = int(abs(work[0]))
    work = zeros((lwork,), t)
    results = lapack_routine(m, n, a, m, tau, work, lwork, 0)

    if results['info'] != 0:
        raise LinAlgError('%s returns %d' % (routine_name, results['info']))

    #  economic mode. Isn't actually economic.
    if mode[0] == 'e':
        if t != result_t :
            a = a.astype(result_t)
        return a.T

    #  generate r
    r = _fastCopyAndTranspose(result_t, a[:,:mn])
    for i in range(mn):
        r[i,:i].fill(0.0)

    #  'r'-mode, that is, calculate only r
    if mode[0] == 'r':
        return r

    #  from here on: build orthonormal matrix q from a

    if isComplexType(t):
        lapack_routine = lapack_lite.zungqr
        routine_name = 'zungqr'
    else:
        lapack_routine = lapack_lite.dorgqr
        routine_name = 'dorgqr'

    # determine optimal lwork
    lwork = 1
    work = zeros((lwork,), t)
    results = lapack_routine(m, mn, mn, a, m, tau, work, -1, 0)
    if results['info'] != 0:
        raise LinAlgError('%s returns %d' % (routine_name, results['info']))

    # compute q
    lwork = int(abs(work[0]))
    work = zeros((lwork,), t)
    results = lapack_routine(m, mn, mn, a, m, tau, work, lwork, 0)
    if results['info'] != 0:
        raise LinAlgError('%s returns %d' % (routine_name, results['info']))

    q = _fastCopyAndTranspose(result_t, a[:mn,:])

    return wrap(q), wrap(r)


# Eigenvalues


def eigvals(a):
    """
    Compute the eigenvalues of a general matrix.

    Main difference between `eigvals` and `eig`: the eigenvectors aren't
    returned.

    Parameters
    ----------
    a : array_like, shape (M, M)
        A complex- or real-valued matrix whose eigenvalues will be computed.

    Returns
    -------
    w : ndarray, shape (M,)
        The eigenvalues, each repeated according to its multiplicity.
        They are not necessarily ordered, nor are they necessarily
        real for real matrices.

    Raises
    ------
    LinAlgError
        If the eigenvalue computation does not converge.

    See Also
    --------
    eig : eigenvalues and right eigenvectors of general arrays
    eigvalsh : eigenvalues of symmetric or Hermitian arrays.
    eigh : eigenvalues and eigenvectors of symmetric/Hermitian arrays.

    Notes
    -----
    This is a simple interface to the LAPACK routines dgeev and zgeev
    that sets those routines' flags to return only the eigenvalues of
    general real and complex arrays, respectively.

    Examples
    --------
    Illustration, using the fact that the eigenvalues of a diagonal matrix
    are its diagonal elements, that multiplying a matrix on the left
    by an orthogonal matrix, `Q`, and on the right by `Q.T` (the transpose
    of `Q`), preserves the eigenvalues of the "middle" matrix.  In other words,
    if `Q` is orthogonal, then ``Q * A * Q.T`` has the same eigenvalues as
    ``A``:

    >>> from numpy import linalg as LA
    >>> x = np.random.random()
    >>> Q = np.array([[np.cos(x), -np.sin(x)], [np.sin(x), np.cos(x)]])
    >>> LA.norm(Q[0, :]), LA.norm(Q[1, :]), np.dot(Q[0, :],Q[1, :])
    (1.0, 1.0, 0.0)

    Now multiply a diagonal matrix by Q on one side and by Q.T on the other:

    >>> D = np.diag((-1,1))
    >>> LA.eigvals(D)
    array([-1.,  1.])
    >>> A = np.dot(Q, D)
    >>> A = np.dot(A, Q.T)
    >>> LA.eigvals(A)
    array([ 1., -1.])

    """
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertSquareness(a)
    _assertFinite(a)
    t, result_t = _commonType(a)
    real_t = _linalgRealType(t)
    a = _fastCopyAndTranspose(t, a)
    a = _to_native_byte_order(a)
    n = a.shape[0]
    dummy = zeros((1,), t)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgeev
        w = zeros((n,), t)
        rwork = zeros((n,), real_t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(_N, _N, n, a, n, w,
                                 dummy, 1, dummy, 1, work, -1, rwork, 0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        results = lapack_routine(_N, _N, n, a, n, w,
                                 dummy, 1, dummy, 1, work, lwork, rwork, 0)
    else:
        lapack_routine = lapack_lite.dgeev
        wr = zeros((n,), t)
        wi = zeros((n,), t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(_N, _N, n, a, n, wr, wi,
                                 dummy, 1, dummy, 1, work, -1, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine(_N, _N, n, a, n, wr, wi,
                                 dummy, 1, dummy, 1, work, lwork, 0)
        if all(wi == 0.):
            w = wr
            result_t = _realType(result_t)
        else:
            w = wr+1j*wi
            result_t = _complexType(result_t)
    if results['info'] > 0:
        raise LinAlgError('Eigenvalues did not converge')
    return w.astype(result_t)


def eigvalsh(a, UPLO='L'):
    """
    Compute the eigenvalues of a Hermitian or real symmetric matrix.

    Main difference from eigh: the eigenvectors are not computed.

    Parameters
    ----------
    a : array_like, shape (M, M)
        A complex- or real-valued matrix whose eigenvalues are to be
        computed.
    UPLO : {'L', 'U'}, optional
        Specifies whether the calculation is done with the lower triangular
        part of `a` ('L', default) or the upper triangular part ('U').

    Returns
    -------
    w : ndarray, shape (M,)
        The eigenvalues, not necessarily ordered, each repeated according to
        its multiplicity.

    Raises
    ------
    LinAlgError
        If the eigenvalue computation does not converge.

    See Also
    --------
    eigh : eigenvalues and eigenvectors of symmetric/Hermitian arrays.
    eigvals : eigenvalues of general real or complex arrays.
    eig : eigenvalues and right eigenvectors of general real or complex
          arrays.

    Notes
    -----
    This is a simple interface to the LAPACK routines dsyevd and zheevd
    that sets those routines' flags to return only the eigenvalues of
    real symmetric and complex Hermitian arrays, respectively.

    Examples
    --------
    >>> from numpy import linalg as LA
    >>> a = np.array([[1, -2j], [2j, 5]])
    >>> LA.eigvalsh(a)
    array([ 0.17157288+0.j,  5.82842712+0.j])

    """
    UPLO = asbytes(UPLO)
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    real_t = _linalgRealType(t)
    a = _fastCopyAndTranspose(t, a)
    a = _to_native_byte_order(a)
    n = a.shape[0]
    liwork = 5*n+3
    iwork = zeros((liwork,), fortran_int)
    if isComplexType(t):
        lapack_routine = lapack_lite.zheevd
        w = zeros((n,), real_t)
        lwork = 1
        work = zeros((lwork,), t)
        lrwork = 1
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine(_N, UPLO, n, a, n, w, work, -1,
                                 rwork, -1, iwork, liwork,  0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        lrwork = int(rwork[0])
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine(_N, UPLO, n, a, n, w, work, lwork,
                                rwork, lrwork, iwork, liwork,  0)
    else:
        lapack_routine = lapack_lite.dsyevd
        w = zeros((n,), t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(_N, UPLO, n, a, n, w, work, -1,
                                 iwork, liwork, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine(_N, UPLO, n, a, n, w, work, lwork,
                                 iwork, liwork, 0)
    if results['info'] > 0:
        raise LinAlgError('Eigenvalues did not converge')
    return w.astype(result_t)

def _convertarray(a):
    t, result_t = _commonType(a)
    a = _fastCT(a.astype(t))
    return a, t, result_t


# Eigenvectors


def eig(a):
    """
    Compute the eigenvalues and right eigenvectors of a square array.

    Parameters
    ----------
    a : array_like, shape (M, M)
        A square array of real or complex elements.

    Returns
    -------
    w : ndarray, shape (M,)
        The eigenvalues, each repeated according to its multiplicity.
        The eigenvalues are not necessarily ordered, nor are they
        necessarily real for real arrays (though for real arrays
        complex-valued eigenvalues should occur in conjugate pairs).

    v : ndarray, shape (M, M)
        The normalized (unit "length") eigenvectors, such that the
        column ``v[:,i]`` is the eigenvector corresponding to the
        eigenvalue ``w[i]``.

    Raises
    ------
    LinAlgError
        If the eigenvalue computation does not converge.

    See Also
    --------
    eigvalsh : eigenvalues of a symmetric or Hermitian (conjugate symmetric)
       array.

    eigvals : eigenvalues of a non-symmetric array.

    Notes
    -----
    This is a simple interface to the LAPACK routines dgeev and zgeev
    which compute the eigenvalues and eigenvectors of, respectively,
    general real- and complex-valued square arrays.

    The number `w` is an eigenvalue of `a` if there exists a vector
    `v` such that ``dot(a,v) = w * v``. Thus, the arrays `a`, `w`, and
    `v` satisfy the equations ``dot(a[i,:], v[i]) = w[i] * v[:,i]``
    for :math:`i \\in \\{0,...,M-1\\}`.

    The array `v` of eigenvectors may not be of maximum rank, that is, some
    of the columns may be linearly dependent, although round-off error may
    obscure that fact. If the eigenvalues are all different, then theoretically
    the eigenvectors are linearly independent. Likewise, the (complex-valued)
    matrix of eigenvectors `v` is unitary if the matrix `a` is normal, i.e.,
    if ``dot(a, a.H) = dot(a.H, a)``, where `a.H` denotes the conjugate
    transpose of `a`.

    Finally, it is emphasized that `v` consists of the *right* (as in
    right-hand side) eigenvectors of `a`.  A vector `y` satisfying
    ``dot(y.T, a) = z * y.T`` for some number `z` is called a *left*
    eigenvector of `a`, and, in general, the left and right eigenvectors
    of a matrix are not necessarily the (perhaps conjugate) transposes
    of each other.

    References
    ----------
    G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando, FL,
    Academic Press, Inc., 1980, Various pp.

    Examples
    --------
    >>> from numpy import linalg as LA

    (Almost) trivial example with real e-values and e-vectors.

    >>> w, v = LA.eig(np.diag((1, 2, 3)))
    >>> w; v
    array([ 1.,  2.,  3.])
    array([[ 1.,  0.,  0.],
           [ 0.,  1.,  0.],
           [ 0.,  0.,  1.]])

    Real matrix possessing complex e-values and e-vectors; note that the
    e-values are complex conjugates of each other.

    >>> w, v = LA.eig(np.array([[1, -1], [1, 1]]))
    >>> w; v
    array([ 1. + 1.j,  1. - 1.j])
    array([[ 0.70710678+0.j        ,  0.70710678+0.j        ],
           [ 0.00000000-0.70710678j,  0.00000000+0.70710678j]])

    Complex-valued matrix with real e-values (but complex-valued e-vectors);
    note that a.conj().T = a, i.e., a is Hermitian.

    >>> a = np.array([[1, 1j], [-1j, 1]])
    >>> w, v = LA.eig(a)
    >>> w; v
    array([  2.00000000e+00+0.j,   5.98651912e-36+0.j]) # i.e., {2, 0}
    array([[ 0.00000000+0.70710678j,  0.70710678+0.j        ],
           [ 0.70710678+0.j        ,  0.00000000+0.70710678j]])

    Be careful about round-off error!

    >>> a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]])
    >>> # Theor. e-values are 1 +/- 1e-9
    >>> w, v = LA.eig(a)
    >>> w; v
    array([ 1.,  1.])
    array([[ 1.,  0.],
           [ 0.,  1.]])

    """
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertSquareness(a)
    _assertFinite(a)
    a, t, result_t = _convertarray(a) # convert to double or cdouble type
    a = _to_native_byte_order(a)
    real_t = _linalgRealType(t)
    n = a.shape[0]
    dummy = zeros((1,), t)
    if isComplexType(t):
        # Complex routines take different arguments
        lapack_routine = lapack_lite.zgeev
        w = zeros((n,), t)
        v = zeros((n, n), t)
        lwork = 1
        work = zeros((lwork,), t)
        rwork = zeros((2*n,), real_t)
        results = lapack_routine(_N, _V, n, a, n, w,
                                 dummy, 1, v, n, work, -1, rwork, 0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        results = lapack_routine(_N, _V, n, a, n, w,
                                 dummy, 1, v, n, work, lwork, rwork, 0)
    else:
        lapack_routine = lapack_lite.dgeev
        wr = zeros((n,), t)
        wi = zeros((n,), t)
        vr = zeros((n, n), t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(_N, _V, n, a, n, wr, wi,
                                  dummy, 1, vr, n, work, -1, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine(_N, _V, n, a, n, wr, wi,
                                  dummy, 1, vr, n, work, lwork, 0)
        if all(wi == 0.0):
            w = wr
            v = vr
            result_t = _realType(result_t)
        else:
            w = wr+1j*wi
            v = array(vr, w.dtype)
            ind = flatnonzero(wi != 0.0)      # indices of complex e-vals
            for i in range(len(ind)//2):
                v[ind[2*i]] = vr[ind[2*i]] + 1j*vr[ind[2*i+1]]
                v[ind[2*i+1]] = vr[ind[2*i]] - 1j*vr[ind[2*i+1]]
            result_t = _complexType(result_t)

    if results['info'] > 0:
        raise LinAlgError('Eigenvalues did not converge')
    vt = v.transpose().astype(result_t)
    return w.astype(result_t), wrap(vt)


def eigh(a, UPLO='L'):
    """
    Return the eigenvalues and eigenvectors of a Hermitian or symmetric matrix.

    Returns two objects, a 1-D array containing the eigenvalues of `a`, and
    a 2-D square array or matrix (depending on the input type) of the
    corresponding eigenvectors (in columns).

    Parameters
    ----------
    a : array_like, shape (M, M)
        A complex Hermitian or real symmetric matrix.
    UPLO : {'L', 'U'}, optional
        Specifies whether the calculation is done with the lower triangular
        part of `a` ('L', default) or the upper triangular part ('U').

    Returns
    -------
    w : ndarray, shape (M,)
        The eigenvalues, not necessarily ordered.
    v : ndarray, or matrix object if `a` is, shape (M, M)
        The column ``v[:, i]`` is the normalized eigenvector corresponding
        to the eigenvalue ``w[i]``.

    Raises
    ------
    LinAlgError
        If the eigenvalue computation does not converge.

    See Also
    --------
    eigvalsh : eigenvalues of symmetric or Hermitian arrays.
    eig : eigenvalues and right eigenvectors for non-symmetric arrays.
    eigvals : eigenvalues of non-symmetric arrays.

    Notes
    -----
    This is a simple interface to the LAPACK routines dsyevd and zheevd,
    which compute the eigenvalues and eigenvectors of real symmetric and
    complex Hermitian arrays, respectively.

    The eigenvalues of real symmetric or complex Hermitian matrices are
    always real. [1]_ The array `v` of (column) eigenvectors is unitary
    and `a`, `w`, and `v` satisfy the equations
    ``dot(a, v[:, i]) = w[i] * v[:, i]``.

    References
    ----------
    .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
           FL, Academic Press, Inc., 1980, pg. 222.

    Examples
    --------
    >>> from numpy import linalg as LA
    >>> a = np.array([[1, -2j], [2j, 5]])
    >>> a
    array([[ 1.+0.j,  0.-2.j],
           [ 0.+2.j,  5.+0.j]])
    >>> w, v = LA.eigh(a)
    >>> w; v
    array([ 0.17157288,  5.82842712])
    array([[-0.92387953+0.j        , -0.38268343+0.j        ],
           [ 0.00000000+0.38268343j,  0.00000000-0.92387953j]])

    >>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair
    array([2.77555756e-17 + 0.j, 0. + 1.38777878e-16j])
    >>> np.dot(a, v[:, 1]) - w[1] * v[:, 1] # verify 2nd e-val/vec pair
    array([ 0.+0.j,  0.+0.j])

    >>> A = np.matrix(a) # what happens if input is a matrix object
    >>> A
    matrix([[ 1.+0.j,  0.-2.j],
            [ 0.+2.j,  5.+0.j]])
    >>> w, v = LA.eigh(A)
    >>> w; v
    array([ 0.17157288,  5.82842712])
    matrix([[-0.92387953+0.j        , -0.38268343+0.j        ],
            [ 0.00000000+0.38268343j,  0.00000000-0.92387953j]])

    """
    UPLO = asbytes(UPLO)
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    real_t = _linalgRealType(t)
    a = _fastCopyAndTranspose(t, a)
    a = _to_native_byte_order(a)
    n = a.shape[0]
    liwork = 5*n+3
    iwork = zeros((liwork,), fortran_int)
    if isComplexType(t):
        lapack_routine = lapack_lite.zheevd
        w = zeros((n,), real_t)
        lwork = 1
        work = zeros((lwork,), t)
        lrwork = 1
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine(_V, UPLO, n, a, n, w, work, -1,
                                 rwork, -1, iwork, liwork,  0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        lrwork = int(rwork[0])
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine(_V, UPLO, n, a, n, w, work, lwork,
                                 rwork, lrwork, iwork, liwork,  0)
    else:
        lapack_routine = lapack_lite.dsyevd
        w = zeros((n,), t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(_V, UPLO, n, a, n, w, work, -1,
                iwork, liwork, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine(_V, UPLO, n, a, n, w, work, lwork,
                iwork, liwork, 0)
    if results['info'] > 0:
        raise LinAlgError('Eigenvalues did not converge')
    at = a.transpose().astype(result_t)
    return w.astype(_realType(result_t)), wrap(at)


# Singular value decomposition

def svd(a, full_matrices=1, compute_uv=1):
    """
    Singular Value Decomposition.

    Factors the matrix `a` as ``u * np.diag(s) * v``, where `u` and `v`
    are unitary and `s` is a 1-d array of `a`'s singular values.

    Parameters
    ----------
    a : array_like
        A real or complex matrix of shape (`M`, `N`) .
    full_matrices : bool, optional
        If True (default), `u` and `v` have the shapes (`M`, `M`) and
        (`N`, `N`), respectively.  Otherwise, the shapes are (`M`, `K`)
        and (`K`, `N`), respectively, where `K` = min(`M`, `N`).
    compute_uv : bool, optional
        Whether or not to compute `u` and `v` in addition to `s`.  True
        by default.

    Returns
    -------
    u : ndarray
        Unitary matrix.  The shape of `u` is (`M`, `M`) or (`M`, `K`)
        depending on value of ``full_matrices``.
    s : ndarray
        The singular values, sorted so that ``s[i] >= s[i+1]``.  `s` is
        a 1-d array of length min(`M`, `N`).
    v : ndarray
        Unitary matrix of shape (`N`, `N`) or (`K`, `N`), depending on
        ``full_matrices``.

    Raises
    ------
    LinAlgError
        If SVD computation does not converge.

    Notes
    -----
    The SVD is commonly written as ``a = U S V.H``.  The `v` returned
    by this function is ``V.H`` and ``u = U``.

    If ``U`` is a unitary matrix, it means that it
    satisfies ``U.H = inv(U)``.

    The rows of `v` are the eigenvectors of ``a.H a``. The columns
    of `u` are the eigenvectors of ``a a.H``.  For row ``i`` in
    `v` and column ``i`` in `u`, the corresponding eigenvalue is
    ``s[i]**2``.

    If `a` is a `matrix` object (as opposed to an `ndarray`), then so
    are all the return values.

    Examples
    --------
    >>> a = np.random.randn(9, 6) + 1j*np.random.randn(9, 6)

    Reconstruction based on full SVD:

    >>> U, s, V = np.linalg.svd(a, full_matrices=True)
    >>> U.shape, V.shape, s.shape
    ((9, 6), (6, 6), (6,))
    >>> S = np.zeros((9, 6), dtype=complex)
    >>> S[:6, :6] = np.diag(s)
    >>> np.allclose(a, np.dot(U, np.dot(S, V)))
    True

    Reconstruction based on reduced SVD:

    >>> U, s, V = np.linalg.svd(a, full_matrices=False)
    >>> U.shape, V.shape, s.shape
    ((9, 6), (6, 6), (6,))
    >>> S = np.diag(s)
    >>> np.allclose(a, np.dot(U, np.dot(S, V)))
    True

    """
    a, wrap = _makearray(a)
    _assertRank2(a)
    _assertNonEmpty(a)
    m, n = a.shape
    t, result_t = _commonType(a)
    real_t = _linalgRealType(t)
    a = _fastCopyAndTranspose(t, a)
    a = _to_native_byte_order(a)
    s = zeros((min(n, m),), real_t)
    if compute_uv:
        if full_matrices:
            nu = m
            nvt = n
            option = _A
        else:
            nu = min(n, m)
            nvt = min(n, m)
            option = _S
        u = zeros((nu, m), t)
        vt = zeros((n, nvt), t)
    else:
        option = _N
        nu = 1
        nvt = 1
        u = empty((1, 1), t)
        vt = empty((1, 1), t)

    iwork = zeros((8*min(m, n),), fortran_int)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgesdd
        lrwork = min(m,n)*max(5*min(m,n)+7, 2*max(m,n)+2*min(m,n)+1)
        rwork = zeros((lrwork,), real_t)
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                 work, -1, rwork, iwork, 0)
        lwork = int(abs(work[0]))
        work = zeros((lwork,), t)
        results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                 work, lwork, rwork, iwork, 0)
    else:
        lapack_routine = lapack_lite.dgesdd
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                 work, -1, iwork, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine(option, m, n, a, m, s, u, m, vt, nvt,
                                 work, lwork, iwork, 0)
    if results['info'] > 0:
        raise LinAlgError('SVD did not converge')
    s = s.astype(_realType(result_t))
    if compute_uv:
        u = u.transpose().astype(result_t)
        vt = vt.transpose().astype(result_t)
        return wrap(u), s, wrap(vt)
    else:
        return s

def cond(x, p=None):
    """
    Compute the condition number of a matrix.

    This function is capable of returning the condition number using
    one of seven different norms, depending on the value of `p` (see
    Parameters below).

    Parameters
    ----------
    x : array_like, shape (M, N)
        The matrix whose condition number is sought.
    p : {None, 1, -1, 2, -2, inf, -inf, 'fro'}, optional
        Order of the norm:

        =====  ============================
        p      norm for matrices
        =====  ============================
        None   2-norm, computed directly using the ``SVD``
        'fro'  Frobenius norm
        inf    max(sum(abs(x), axis=1))
        -inf   min(sum(abs(x), axis=1))
        1      max(sum(abs(x), axis=0))
        -1     min(sum(abs(x), axis=0))
        2      2-norm (largest sing. value)
        -2     smallest singular value
        =====  ============================

        inf means the numpy.inf object, and the Frobenius norm is
        the root-of-sum-of-squares norm.

    Returns
    -------
    c : {float, inf}
        The condition number of the matrix. May be infinite.

    See Also
    --------
    numpy.linalg.norm

    Notes
    -----
    The condition number of `x` is defined as the norm of `x` times the
    norm of the inverse of `x` [1]_; the norm can be the usual L2-norm
    (root-of-sum-of-squares) or one of a number of other matrix norms.

    References
    ----------
    .. [1] G. Strang, *Linear Algebra and Its Applications*, Orlando, FL,
           Academic Press, Inc., 1980, pg. 285.

    Examples
    --------
    >>> from numpy import linalg as LA
    >>> a = np.array([[1, 0, -1], [0, 1, 0], [1, 0, 1]])
    >>> a
    array([[ 1,  0, -1],
           [ 0,  1,  0],
           [ 1,  0,  1]])
    >>> LA.cond(a)
    1.4142135623730951
    >>> LA.cond(a, 'fro')
    3.1622776601683795
    >>> LA.cond(a, np.inf)
    2.0
    >>> LA.cond(a, -np.inf)
    1.0
    >>> LA.cond(a, 1)
    2.0
    >>> LA.cond(a, -1)
    1.0
    >>> LA.cond(a, 2)
    1.4142135623730951
    >>> LA.cond(a, -2)
    0.70710678118654746
    >>> min(LA.svd(a, compute_uv=0))*min(LA.svd(LA.inv(a), compute_uv=0))
    0.70710678118654746

    """
    x = asarray(x) # in case we have a matrix
    if p is None:
        s = svd(x,compute_uv=False)
        return s[0]/s[-1]
    else:
        return norm(x,p)*norm(inv(x),p)


def matrix_rank(M, tol=None):
    """
    Return matrix rank of array using SVD method

    Rank of the array is the number of SVD singular values of the
    array that are greater than `tol`.

    Parameters
    ----------
    M : array_like
        array of <=2 dimensions
    tol : {None, float}
       threshold below which SVD values are considered zero. If `tol` is
       None, and ``S`` is an array with singular values for `M`, and
       ``eps`` is the epsilon value for datatype of ``S``, then `tol` is
       set to ``S.max() * eps``.

    Notes
    -----
    Golub and van Loan [1]_ define "numerical rank deficiency" as using
    tol=eps*S[0] (where S[0] is the maximum singular value and thus the
    2-norm of the matrix). This is one definition of rank deficiency,
    and the one we use here.  When floating point roundoff is the main
    concern, then "numerical rank deficiency" is a reasonable choice. In
    some cases you may prefer other definitions. The most useful measure
    of the tolerance depends on the operations you intend to use on your
    matrix. For example, if your data come from uncertain measurements
    with uncertainties greater than floating point epsilon, choosing a
    tolerance near that uncertainty may be preferable.  The tolerance
    may be absolute if the uncertainties are absolute rather than
    relative.

    References
    ----------
    .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*.
       Baltimore: Johns Hopkins University Press, 1996.

    Examples
    --------
    >>> matrix_rank(np.eye(4)) # Full rank matrix
    4
    >>> I=np.eye(4); I[-1,-1] = 0. # rank deficient matrix
    >>> matrix_rank(I)
    3
    >>> matrix_rank(np.ones((4,))) # 1 dimension - rank 1 unless all 0
    1
    >>> matrix_rank(np.zeros((4,)))
    0

    """
    M = asarray(M)
    if M.ndim > 2:
        raise TypeError('array should have 2 or fewer dimensions')
    if M.ndim < 2:
        return int(not all(M==0))
    S = svd(M, compute_uv=False)
    if tol is None:
        tol = S.max() * finfo(S.dtype).eps
    return sum(S > tol)


# Generalized inverse

def pinv(a, rcond=1e-15 ):
    """
    Compute the (Moore-Penrose) pseudo-inverse of a matrix.

    Calculate the generalized inverse of a matrix using its
    singular-value decomposition (SVD) and including all
    *large* singular values.

    Parameters
    ----------
    a : array_like, shape (M, N)
      Matrix to be pseudo-inverted.
    rcond : float
      Cutoff for small singular values.
      Singular values smaller (in modulus) than
      `rcond` * largest_singular_value (again, in modulus)
      are set to zero.

    Returns
    -------
    B : ndarray, shape (N, M)
      The pseudo-inverse of `a`. If `a` is a `matrix` instance, then so
      is `B`.

    Raises
    ------
    LinAlgError
      If the SVD computation does not converge.

    Notes
    -----
    The pseudo-inverse of a matrix A, denoted :math:`A^+`, is
    defined as: "the matrix that 'solves' [the least-squares problem]
    :math:`Ax = b`," i.e., if :math:`\\bar{x}` is said solution, then
    :math:`A^+` is that matrix such that :math:`\\bar{x} = A^+b`.

    It can be shown that if :math:`Q_1 \\Sigma Q_2^T = A` is the singular
    value decomposition of A, then
    :math:`A^+ = Q_2 \\Sigma^+ Q_1^T`, where :math:`Q_{1,2}` are
    orthogonal matrices, :math:`\\Sigma` is a diagonal matrix consisting
    of A's so-called singular values, (followed, typically, by
    zeros), and then :math:`\\Sigma^+` is simply the diagonal matrix
    consisting of the reciprocals of A's singular values
    (again, followed by zeros). [1]_

    References
    ----------
    .. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
           FL, Academic Press, Inc., 1980, pp. 139-142.

    Examples
    --------
    The following example checks that ``a * a+ * a == a`` and
    ``a+ * a * a+ == a+``:

    >>> a = np.random.randn(9, 6)
    >>> B = np.linalg.pinv(a)
    >>> np.allclose(a, np.dot(a, np.dot(B, a)))
    True
    >>> np.allclose(B, np.dot(B, np.dot(a, B)))
    True

    """
    a, wrap = _makearray(a)
    _assertNonEmpty(a)
    a = a.conjugate()
    u, s, vt = svd(a, 0)
    m = u.shape[0]
    n = vt.shape[1]
    cutoff = rcond*maximum.reduce(s)
    for i in range(min(n, m)):
        if s[i] > cutoff:
            s[i] = 1./s[i]
        else:
            s[i] = 0.;
    res = dot(transpose(vt), multiply(s[:, newaxis],transpose(u)))
    return wrap(res)

# Determinant

def slogdet(a):
    """
    Compute the sign and (natural) logarithm of the determinant of an array.

    If an array has a very small or very large determinant, than a call to
    `det` may overflow or underflow. This routine is more robust against such
    issues, because it computes the logarithm of the determinant rather than
    the determinant itself.

    Parameters
    ----------
    a : array_like
        Input array, has to be a square 2-D array.

    Returns
    -------
    sign : float or complex
        A number representing the sign of the determinant. For a real matrix,
        this is 1, 0, or -1. For a complex matrix, this is a complex number
        with absolute value 1 (i.e., it is on the unit circle), or else 0.
    logdet : float
        The natural log of the absolute value of the determinant.

    If the determinant is zero, then `sign` will be 0 and `logdet` will be
    -Inf. In all cases, the determinant is equal to ``sign * np.exp(logdet)``.

    See Also
    --------
    det

    Notes
    -----
    The determinant is computed via LU factorization using the LAPACK
    routine z/dgetrf.

    .. versionadded:: 1.6.0.

    Examples
    --------
    The determinant of a 2-D array ``[[a, b], [c, d]]`` is ``ad - bc``:

    >>> a = np.array([[1, 2], [3, 4]])
    >>> (sign, logdet) = np.linalg.slogdet(a)
    >>> (sign, logdet)
    (-1, 0.69314718055994529)
    >>> sign * np.exp(logdet)
    -2.0

    This routine succeeds where ordinary `det` does not:

    >>> np.linalg.det(np.eye(500) * 0.1)
    0.0
    >>> np.linalg.slogdet(np.eye(500) * 0.1)
    (1, -1151.2925464970228)

    """
    a = asarray(a)
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    a = _fastCopyAndTranspose(t, a)
    a = _to_native_byte_order(a)
    n = a.shape[0]
    if isComplexType(t):
        lapack_routine = lapack_lite.zgetrf
    else:
        lapack_routine = lapack_lite.dgetrf
    pivots = zeros((n,), fortran_int)
    results = lapack_routine(n, n, a, n, pivots, 0)
    info = results['info']
    if (info < 0):
        raise TypeError("Illegal input to Fortran routine")
    elif (info > 0):
        return (t(0.0), _realType(t)(-Inf))
    sign = 1. - 2. * (add.reduce(pivots != arange(1, n + 1)) % 2)
    d = diagonal(a)
    absd = absolute(d)
    sign *= multiply.reduce(d / absd)
    log(absd, absd)
    logdet = add.reduce(absd, axis=-1)
    return sign, logdet

def det(a):
    """
    Compute the determinant of an array.

    Parameters
    ----------
    a : array_like, shape (M, M)
        Input array.

    Returns
    -------
    det : ndarray
        Determinant of `a`.

    Notes
    -----
    The determinant is computed via LU factorization using the LAPACK
    routine z/dgetrf.

    Examples
    --------
    The determinant of a 2-D array [[a, b], [c, d]] is ad - bc:

    >>> a = np.array([[1, 2], [3, 4]])
    >>> np.linalg.det(a)
    -2.0

    See Also
    --------
    slogdet : Another way to representing the determinant, more suitable
      for large matrices where underflow/overflow may occur.

    """
    sign, logdet = slogdet(a)
    return sign * exp(logdet)

# Linear Least Squares

def lstsq(a, b, rcond=-1):
    """
    Return the least-squares solution to a linear matrix equation.

    Solves the equation `a x = b` by computing a vector `x` that
    minimizes the Euclidean 2-norm `|| b - a x ||^2`.  The equation may
    be under-, well-, or over- determined (i.e., the number of
    linearly independent rows of `a` can be less than, equal to, or
    greater than its number of linearly independent columns).  If `a`
    is square and of full rank, then `x` (but for round-off error) is
    the "exact" solution of the equation.

    Parameters
    ----------
    a : array_like, shape (M, N)
        "Coefficient" matrix.
    b : array_like, shape (M,) or (M, K)
        Ordinate or "dependent variable" values. If `b` is two-dimensional,
        the least-squares solution is calculated for each of the `K` columns
        of `b`.
    rcond : float, optional
        Cut-off ratio for small singular values of `a`.
        Singular values are set to zero if they are smaller than `rcond`
        times the largest singular value of `a`.

    Returns
    -------
    x : ndarray, shape (N,) or (N, K)
        Least-squares solution.  The shape of `x` depends on the shape of
        `b`.
    residues : ndarray, shape (), (1,), or (K,)
        Sums of residues; squared Euclidean 2-norm for each column in
        ``b - a*x``.
        If the rank of `a` is < N or > M, this is an empty array.
        If `b` is 1-dimensional, this is a (1,) shape array.
        Otherwise the shape is (K,).
    rank : int
        Rank of matrix `a`.
    s : ndarray, shape (min(M,N),)
        Singular values of `a`.

    Raises
    ------
    LinAlgError
        If computation does not converge.

    Notes
    -----
    If `b` is a matrix, then all array results are returned as matrices.

    Examples
    --------
    Fit a line, ``y = mx + c``, through some noisy data-points:

    >>> x = np.array([0, 1, 2, 3])
    >>> y = np.array([-1, 0.2, 0.9, 2.1])

    By examining the coefficients, we see that the line should have a
    gradient of roughly 1 and cut the y-axis at, more or less, -1.

    We can rewrite the line equation as ``y = Ap``, where ``A = [[x 1]]``
    and ``p = [[m], [c]]``.  Now use `lstsq` to solve for `p`:

    >>> A = np.vstack([x, np.ones(len(x))]).T
    >>> A
    array([[ 0.,  1.],
           [ 1.,  1.],
           [ 2.,  1.],
           [ 3.,  1.]])

    >>> m, c = np.linalg.lstsq(A, y)[0]
    >>> print m, c
    1.0 -0.95

    Plot the data along with the fitted line:

    >>> import matplotlib.pyplot as plt
    >>> plt.plot(x, y, 'o', label='Original data', markersize=10)
    >>> plt.plot(x, m*x + c, 'r', label='Fitted line')
    >>> plt.legend()
    >>> plt.show()

    """
    import math
    a, _ = _makearray(a)
    b, wrap = _makearray(b)
    is_1d = len(b.shape) == 1
    if is_1d:
        b = b[:, newaxis]
    _assertRank2(a, b)
    m  = a.shape[0]
    n  = a.shape[1]
    n_rhs = b.shape[1]
    ldb = max(n, m)
    if m != b.shape[0]:
        raise LinAlgError('Incompatible dimensions')
    t, result_t = _commonType(a, b)
    result_real_t = _realType(result_t)
    real_t = _linalgRealType(t)
    bstar = zeros((ldb, n_rhs), t)
    bstar[:b.shape[0],:n_rhs] = b.copy()
    a, bstar = _fastCopyAndTranspose(t, a, bstar)
    a, bstar = _to_native_byte_order(a, bstar)
    s = zeros((min(m, n),), real_t)
    nlvl = max( 0, int( math.log( float(min(m, n))/2. ) ) + 1 )
    iwork = zeros((3*min(m, n)*nlvl+11*min(m, n),), fortran_int)
    if isComplexType(t):
        lapack_routine = lapack_lite.zgelsd
        lwork = 1
        rwork = zeros((lwork,), real_t)
        work = zeros((lwork,), t)
        results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                 0, work, -1, rwork, iwork, 0)
        lwork = int(abs(work[0]))
        rwork = zeros((lwork,), real_t)
        a_real = zeros((m, n), real_t)
        bstar_real = zeros((ldb, n_rhs,), real_t)
        results = lapack_lite.dgelsd(m, n, n_rhs, a_real, m,
                                     bstar_real, ldb, s, rcond,
                                     0, rwork, -1, iwork, 0)
        lrwork = int(rwork[0])
        work = zeros((lwork,), t)
        rwork = zeros((lrwork,), real_t)
        results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                 0, work, lwork, rwork, iwork, 0)
    else:
        lapack_routine = lapack_lite.dgelsd
        lwork = 1
        work = zeros((lwork,), t)
        results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                 0, work, -1, iwork, 0)
        lwork = int(work[0])
        work = zeros((lwork,), t)
        results = lapack_routine(m, n, n_rhs, a, m, bstar, ldb, s, rcond,
                                 0, work, lwork, iwork, 0)
    if results['info'] > 0:
        raise LinAlgError('SVD did not converge in Linear Least Squares')
    resids = array([], result_real_t)
    if is_1d:
        x = array(ravel(bstar)[:n], dtype=result_t, copy=True)
        if results['rank'] == n and m > n:
            if isComplexType(t):
                resids = array([sum(abs(ravel(bstar)[n:])**2)],
                               dtype=result_real_t)
            else:
                resids = array([sum((ravel(bstar)[n:])**2)],
                               dtype=result_real_t)
    else:
        x = array(transpose(bstar)[:n,:], dtype=result_t, copy=True)
        if results['rank'] == n and m > n:
            if isComplexType(t):
                resids = sum(abs(transpose(bstar)[n:,:])**2, axis=0).astype(
                    result_real_t)
            else:
                resids = sum((transpose(bstar)[n:,:])**2, axis=0).astype(
                    result_real_t)

    st = s[:min(n, m)].copy().astype(result_real_t)
    return wrap(x), wrap(resids), results['rank'], st

def norm(x, ord=None):
    """
    Matrix or vector norm.

    This function is able to return one of seven different matrix norms,
    or one of an infinite number of vector norms (described below), depending
    on the value of the ``ord`` parameter.

    Parameters
    ----------
    x : array_like, shape (M,) or (M, N)
        Input array.
    ord : {non-zero int, inf, -inf, 'fro'}, optional
        Order of the norm (see table under ``Notes``). inf means numpy's
        `inf` object.

    Returns
    -------
    n : float
        Norm of the matrix or vector.

    Notes
    -----
    For values of ``ord <= 0``, the result is, strictly speaking, not a
    mathematical 'norm', but it may still be useful for various numerical
    purposes.

    The following norms can be calculated:

    =====  ============================  ==========================
    ord    norm for matrices             norm for vectors
    =====  ============================  ==========================
    None   Frobenius norm                2-norm
    'fro'  Frobenius norm                --
    inf    max(sum(abs(x), axis=1))      max(abs(x))
    -inf   min(sum(abs(x), axis=1))      min(abs(x))
    0      --                            sum(x != 0)
    1      max(sum(abs(x), axis=0))      as below
    -1     min(sum(abs(x), axis=0))      as below
    2      2-norm (largest sing. value)  as below
    -2     smallest singular value       as below
    other  --                            sum(abs(x)**ord)**(1./ord)
    =====  ============================  ==========================

    The Frobenius norm is given by [1]_:

        :math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}`

    References
    ----------
    .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
           Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

    Examples
    --------
    >>> from numpy import linalg as LA
    >>> a = np.arange(9) - 4
    >>> a
    array([-4, -3, -2, -1,  0,  1,  2,  3,  4])
    >>> b = a.reshape((3, 3))
    >>> b
    array([[-4, -3, -2],
           [-1,  0,  1],
           [ 2,  3,  4]])

    >>> LA.norm(a)
    7.745966692414834
    >>> LA.norm(b)
    7.745966692414834
    >>> LA.norm(b, 'fro')
    7.745966692414834
    >>> LA.norm(a, np.inf)
    4
    >>> LA.norm(b, np.inf)
    9
    >>> LA.norm(a, -np.inf)
    0
    >>> LA.norm(b, -np.inf)
    2

    >>> LA.norm(a, 1)
    20
    >>> LA.norm(b, 1)
    7
    >>> LA.norm(a, -1)
    -4.6566128774142013e-010
    >>> LA.norm(b, -1)
    6
    >>> LA.norm(a, 2)
    7.745966692414834
    >>> LA.norm(b, 2)
    7.3484692283495345

    >>> LA.norm(a, -2)
    nan
    >>> LA.norm(b, -2)
    1.8570331885190563e-016
    >>> LA.norm(a, 3)
    5.8480354764257312
    >>> LA.norm(a, -3)
    nan

    """
    x = asarray(x)
    if ord is None: # check the default case first and handle it immediately
        return sqrt(add.reduce((x.conj() * x).ravel().real))

    nd = x.ndim
    if nd == 1:
        if ord == Inf:
            return abs(x).max()
        elif ord == -Inf:
            return abs(x).min()
        elif ord == 0:
            return (x != 0).sum() # Zero norm
        elif ord == 1:
            return abs(x).sum() # special case for speedup
        elif ord == 2:
            return sqrt(((x.conj()*x).real).sum()) # special case for speedup
        else:
            try:
                ord + 1
            except TypeError:
                raise ValueError("Invalid norm order for vectors.")
            return ((abs(x)**ord).sum())**(1.0/ord)
    elif nd == 2:
        if ord == 2:
            return svd(x, compute_uv=0).max()
        elif ord == -2:
            return svd(x, compute_uv=0).min()
        elif ord == 1:
            return abs(x).sum(axis=0).max()
        elif ord == Inf:
            return abs(x).sum(axis=1).max()
        elif ord == -1:
            return abs(x).sum(axis=0).min()
        elif ord == -Inf:
            return abs(x).sum(axis=1).min()
        elif ord in ['fro','f']:
            return sqrt(add.reduce((x.conj() * x).real.ravel()))
        else:
            raise ValueError("Invalid norm order for matrices.")
    else:
        raise ValueError("Improper number of dimensions to norm.")