This file is indexed.

/usr/include/trilinos/BelosOrthoManagerTest.hpp is in libtrilinos-belos-dev 12.10.1-3.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

   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
//@HEADER
// ************************************************************************
//
//                 Belos: Block Linear Solvers Package
//                  Copyright 2004 Sandia Corporation
//
// Under the terms of Contract DE-AC04-94AL85000 with Sandia Corporation,
// the U.S. Government retains certain rights in this software.
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
// 1. Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
//
// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
//
// 3. Neither the name of the Corporation nor the names of the
// contributors may be used to endorse or promote products derived from
// this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY SANDIA CORPORATION "AS IS" AND ANY
// EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
// PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL SANDIA CORPORATION OR THE
// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
// LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
// Questions? Contact Michael A. Heroux (maherou@sandia.gov)
//
// ************************************************************************
//@HEADER

/// \file BelosOrthoManagerTest.hpp
/// \brief Tests for Belos::OrthoManager and Belos::MatOrthoManager subclasses
///

#include <BelosConfigDefs.hpp>
#include <BelosMultiVecTraits.hpp>
#include <BelosOutputManager.hpp>
#include <BelosOrthoManagerFactory.hpp>
#include <Teuchos_StandardCatchMacros.hpp>
#include <Teuchos_TimeMonitor.hpp>
#include <iostream>
#include <stdexcept>

using std::endl;

namespace Belos {
  namespace Test {

    /// \class OrthoManagerBenchmarker
    /// \brief OrthoManager benchmark
    /// \author Mark Hoemmen
    ///
    template<class Scalar, class MV>
    class OrthoManagerBenchmarker {
    private:
      typedef Scalar scalar_type;
      typedef typename Teuchos::ScalarTraits<Scalar>::magnitudeType magnitude_type;
      typedef MultiVecTraits<Scalar, MV> MVT;
      typedef Teuchos::SerialDenseMatrix<int, Scalar> mat_type;

    public:
      /// \brief Establish baseline run time for OrthoManager benchmark
      ///
      /// Replacing a Belos OrthoManager or MatOrthoManager's
      /// projection and normalization operations with the same number
      /// of vector copies establishes a rough lower bound on run
      /// time, because orthogonalization generally requires that much
      /// data movement.  This gives us a rough sense for how long the
      /// orthogonalization should take, so we can calibrate the
      /// number of trials needed for accurate timings.
      static void
      baseline (const Teuchos::RCP<const MV>& X,
                const int numCols,
                const int numBlocks,
                const int numTrials)
      {
        using Teuchos::Array;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::Time;
        using Teuchos::TimeMonitor;

        // Make some blocks to "orthogonalize."  Fill with random
        // data.  We only need X so that we can make clones (it knows
        // its data distribution).
        Array<RCP<MV> > V (numBlocks);
        for (int k = 0; k < numBlocks; ++k) {
          V[k] = MVT::Clone (*X, numCols);
          MVT::MvRandom (*V[k]);
        }

        // Make timers with informative labels
        RCP<Time> timer = TimeMonitor::getNewCounter ("Baseline for OrthoManager benchmark");

        // Baseline benchmark just copies data.  It's sort of a lower
        // bound proxy for the volume of data movement done by a real
        // OrthoManager.
        {
          TimeMonitor monitor (*timer);
          for (int trial = 0; trial < numTrials; ++trial) {
            for (int k = 0; k < numBlocks; ++k) {
              for (int j = 0; j < k; ++j)
                MVT::Assign (*V[j], *V[k]);
              MVT::Assign (*X, *V[k]);
            }
          }
        }
      }

      /// \brief Benchmark the given orthogonalization manager
      ///
      /// \param orthoMan [in(/out)] The orthogonalization
      ///   manager to benchmark
      /// \param orthoManName [in] Name of the orthogonalization
      ///   manager (e.g., "TSQR", "ICGS", "DGKS")
      /// \param normalization [in] Normalization scheme used
      ///   by the orthogonalization manager (only applicable
      ///   to the "Simple" orthogonalization)
      /// \param X [in] "Prototype" multivector; not modified
      /// \param numCols [in] Number of columns per block
      /// \param numBlocks [in] Number of blocks
      /// \param numTrials [in] Number of trials in the timing run
      /// \param outMan [out] Output manager
      ///
      /// \param resultStream [out] Output stream for printing
      ///   benchmark results.  If displayResultsCompactly is true, it
      ///   will be written by all MPI rank(s), so on ranks other than
      ///   0, it should be set appropriately to a "black hole stream"
      ///   that doesn't write anything.
      ///
      /// \param displayResultsCompactly [in] If false, rely on
      ///   TimeMonitor::summarize() to print results to resultStream
      ///   (and ensure only MPI Rank 0 does so).  If true, print
      ///   results in a more compact format suitable for automatic
      ///   parsing, using a CSV (Comma-Delimited Values) parser.  In
      ///   "compact" mode, two lines are printed, both of which are
      ///   comma-delimited ASCII text.  The first line begins with a
      ///   "comment" character #; following that are column ("field")
      ///   labels.  The second line contains the actual data, again
      ///   in ASCII comma-delimited format.
      static void
      benchmark (const Teuchos::RCP<OrthoManager<Scalar, MV> >& orthoMan,
                 const std::string& orthoManName,
                 const std::string& normalization,
                 const Teuchos::RCP<const MV>& X,
                 const int numCols,
                 const int numBlocks,
                 const int numTrials,
                 const Teuchos::RCP<OutputManager<Scalar> >& outMan,
                 std::ostream& resultStream,
                 const bool displayResultsCompactly=false)
      {
        using Teuchos::Array;
        using Teuchos::ArrayView;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::Time;
        using Teuchos::TimeMonitor;
        using std::endl;

        TEUCHOS_TEST_FOR_EXCEPTION(orthoMan.is_null(), std::invalid_argument,
                           "orthoMan is null");
        TEUCHOS_TEST_FOR_EXCEPTION(X.is_null(), std::invalid_argument,
                           "X is null");
        TEUCHOS_TEST_FOR_EXCEPTION(numCols < 1, std::invalid_argument,
                           "numCols = " << numCols << " < 1");
        TEUCHOS_TEST_FOR_EXCEPTION(numBlocks < 1, std::invalid_argument,
                           "numBlocks = " << numBlocks << " < 1");
        TEUCHOS_TEST_FOR_EXCEPTION(numTrials < 1, std::invalid_argument,
                           "numTrials = " << numTrials << " < 1");
        // Debug output stream
        std::ostream& debugOut = outMan->stream(Debug);

        // If you like, you can add the "baseline" as an approximate
        // lower bound for orthogonalization performance.  It may be
        // useful as a sanity check to make sure that your
        // orthogonalizations are really computing something, though
        // testing accuracy can help with that too.
        //
        //baseline (X, numCols, numBlocks, numTrials);

        // Make space to put the projection and normalization
        // coefficients.
        Array<RCP<mat_type> > C (numBlocks);
        for (int k = 0; k < numBlocks; ++k) {
          C[k] = rcp (new mat_type (numCols, numCols));
        }
        RCP<mat_type> B (new mat_type (numCols, numCols));

        // Make some blocks to orthogonalize.  Fill with random data.
        // We won't be orthogonalizing X, or even modifying X.  We
        // only need X so that we can make clones (since X knows its
        // data distribution).
        Array<RCP<MV> > V (numBlocks);
        for (int k = 0; k < numBlocks; ++k) {
          V[k] = MVT::Clone (*X, numCols);
          MVT::MvRandom (*V[k]);
        }

        // Make timers with informative labels.  We time an additional
        // first run to measure the startup costs, if any, of the
        // OrthoManager instance.
        RCP<Time> firstRunTimer;
        {
          std::ostringstream os;
          os << "OrthoManager: " << orthoManName << " first run";
          firstRunTimer = TimeMonitor::getNewCounter (os.str());
        }
        RCP<Time> timer;
        {
          std::ostringstream os;
          os << "OrthoManager: " << orthoManName << " total over "
             << numTrials << " trials (excluding first run above)";
          timer = TimeMonitor::getNewCounter (os.str());
        }
        // The first run lets us measure the startup costs, if any, of
        // the OrthoManager instance, without these costs influencing
        // the following timing runs.
        {
          TimeMonitor monitor (*firstRunTimer);
          {
            (void) orthoMan->normalize (*V[0], B);
            for (int k = 1; k < numBlocks; ++k) {
              // k is the number of elements in the ArrayView.  We
              // have to assign first to an ArrayView-of-RCP-of-MV,
              // rather than to an ArrayView-of-RCP-of-const-MV, since
              // the latter requires a reinterpret cast.  Don't you
              // love C++ type inference?
              ArrayView<RCP<MV> > V_0k_nonconst = V.view (0, k);
              ArrayView<RCP<const MV> > V_0k =
                Teuchos::av_reinterpret_cast<RCP<const MV> > (V_0k_nonconst);
              (void) orthoMan->projectAndNormalize (*V[k], C, B, V_0k);
            }
          }
          // "Test" that the trial run actually orthogonalized
          // correctly.  Results are printed to the OutputManager's
          // Belos::Debug output stream, so depending on the
          // OutputManager's chosen verbosity level, you may or may
          // not see the results of the test.
          //
          // NOTE (mfh 22 Jan 2011) For now, these results have to be
          // inspected visually.  We should add a simple automatic
          // test.
          debugOut << "Orthogonality of V[0:" << (numBlocks-1)
                   << "]:" << endl;
          for (int k = 0; k < numBlocks; ++k) {
            // Orthogonality of each block
            debugOut << "For block V[" << k << "]:" << endl;
            debugOut << "  ||<V[" << k << "], V[" << k << "]> - I|| = "
                     << orthoMan->orthonormError(*V[k]) << endl;
            // Relative orthogonality with the previous blocks
            for (int j = 0; j < k; ++j) {
              debugOut << "  ||< V[" << j << "], V[" << k << "] >|| = "
                       << orthoMan->orthogError(*V[j], *V[k]) << endl;
            }
          }
        }

        // Run the benchmark for numTrials trials.  Time all trials as
        // a single run.
        {
          TimeMonitor monitor (*timer);

          for (int trial = 0; trial < numTrials; ++trial) {
            (void) orthoMan->normalize (*V[0], B);
            for (int k = 1; k < numBlocks; ++k) {
              ArrayView<RCP<MV> > V_0k_nonconst = V.view (0, k);
              ArrayView<RCP<const MV> > V_0k =
                Teuchos::av_reinterpret_cast<RCP<const MV> > (V_0k_nonconst);
              (void) orthoMan->projectAndNormalize (*V[k], C, B, V_0k);
            }
          }
        }

        // Report timing results.
        if (displayResultsCompactly)
          {
            // The "compact" format is suitable for automatic parsing,
            // using a CSV (Comma-Delimited Values) parser.  The first
            // "comment" line may be parsed to extract column
            // ("field") labels; the second line contains the actual
            // data, in ASCII comma-delimited format.
            using std::endl;
            resultStream << "#orthoManName"
                         << ",normalization"
                         << ",numRows"
                         << ",numCols"
                         << ",numBlocks"
                         << ",firstRunTimeInSeconds"
                         << ",timeInSeconds"
                         << ",numTrials"
                         << endl;
            resultStream << orthoManName
                         << "," << (orthoManName=="Simple" ? normalization : "N/A")
                         << "," << MVT::GetGlobalLength(*X)
                         << "," << numCols
                         << "," << numBlocks
                         << "," << firstRunTimer->totalElapsedTime()
                         << "," << timer->totalElapsedTime()
                         << "," << numTrials
                         << endl;
          }
        else {
          TimeMonitor::summarize (resultStream);
        }
      }
    };

    /// \class OrthoManagerTester
    /// \brief Wrapper around OrthoManager test functionality
    ///
    template< class Scalar, class MV >
    class OrthoManagerTester {
    private:
      typedef typename Teuchos::Array<Teuchos::RCP<MV> >::size_type size_type;

    public:
      typedef Scalar scalar_type;
      typedef Teuchos::ScalarTraits<scalar_type> SCT;
      typedef typename SCT::magnitudeType magnitude_type;
      typedef Teuchos::ScalarTraits<magnitude_type> SMT;
      typedef MultiVecTraits<scalar_type, MV> MVT;
      typedef Teuchos::SerialDenseMatrix<int, scalar_type> mat_type;

      /// \brief Run all the tests
      ///
      /// \param OM [in/out] OrthoManager subclass instance to test
      /// \param isRankRevealing [in] Whether that OrthoManager
      ///   subclass instance has a true rank-revealing capability.
      ///   If not, we do not test it on rank-deficient vectors.
      /// \param S [in/out] Multivector instance
      /// \param sizeX1 [in] Number of columns in X1 (a multivector
      ///   instance created internally for tests)
      /// \param sizeX2 [in] Number of columns in X2 (a multivector
      ///   instance created internally for tests)
      /// \param MyOM [out] Output manager for handling local output.
      ///   In Anasazi, this class is called BasicOutputManager.  In
      ///   Belos, this class is called OutputManager.
      ///
      /// \return Number of tests that failed (zero means success)
      static int
      runTests (const Teuchos::RCP<OrthoManager<Scalar, MV> >& OM,
                const bool isRankRevealing,
                const Teuchos::RCP<MV>& S,
                const int sizeX1,
                const int sizeX2,
                const Teuchos::RCP<OutputManager<Scalar> >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::rcp_dynamic_cast;
        using Teuchos::tuple;

        // Number of tests that have failed thus far.
        int numFailed = 0;

        // Relative tolerance against which all tests are performed.
        const magnitude_type TOL = 1.0e-12;
        // Absolute tolerance constant
        //const magnitude_type ATOL = 10;

        const scalar_type ZERO = SCT::zero();
        const scalar_type ONE = SCT::one();

        // Debug output stream
        std::ostream& debugOut = MyOM->stream(Debug);

        // Number of columns in the input "prototype" multivector S.
        const int sizeS = MVT::GetNumberVecs (*S);

        // Create multivectors X1 and X2, using the same map as multivector
        // S.  Then, test orthogonalizing X2 against X1.  After doing so, X1
        // and X2 should each be M-orthonormal, and should be mutually
        // M-orthogonal.
        debugOut << "Generating X1,X2 for testing... ";
        RCP< MV > X1 = MVT::Clone (*S, sizeX1);
        RCP< MV > X2 = MVT::Clone (*S, sizeX2);
        debugOut << "done." << endl;
        {
          magnitude_type err;

          //
          // Fill X1 with random values, and test the normalization error.
          //
          debugOut << "Filling X1 with random values... ";
          MVT::MvRandom(*X1);
          debugOut << "done." << endl
                   << "Calling normalize() on X1... ";
          // The Anasazi and Belos OrthoManager interfaces differ.
          // For example, Anasazi's normalize() method accepts either
          // one or two arguments, whereas Belos' normalize() requires
          // two arguments.
          const int initialX1Rank = OM->normalize(*X1, Teuchos::null);
          TEUCHOS_TEST_FOR_EXCEPTION(initialX1Rank != sizeX1,
                             std::runtime_error,
                             "normalize(X1) returned rank "
                             << initialX1Rank << " from " << sizeX1
                             << " vectors. Cannot continue.");
          debugOut << "done." << endl
                   << "Calling orthonormError() on X1... ";
          err = OM->orthonormError(*X1);
          TEUCHOS_TEST_FOR_EXCEPTION(err > TOL, std::runtime_error,
                             "After normalize(X1), orthonormError(X1) = "
                             << err << " > TOL = " << TOL);
          debugOut << "done: ||<X1,X1> - I|| = " << err << endl;

          //
          // Fill X2 with random values, project against X1 and normalize,
          // and test the orthogonalization error.
          //
          debugOut << "Filling X2 with random values... ";
          MVT::MvRandom(*X2);
          debugOut << "done." << endl
                   << "Calling projectAndNormalize(X2, C, B, tuple(X1))... "
                   << std::flush;
          // The projectAndNormalize() interface also differs between
          // Anasazi and Belos.  Anasazi's projectAndNormalize() puts
          // the multivector and the array of multivectors first, and
          // the (array of) SerialDenseMatrix arguments (which are
          // optional) afterwards.  Belos puts the (array of)
          // SerialDenseMatrix arguments in the middle, and they are
          // not optional.
          int initialX2Rank;
          {
            Array<RCP<mat_type> > C (1);
            RCP<mat_type> B = Teuchos::null;
            initialX2Rank =
              OM->projectAndNormalize (*X2, C, B, tuple<RCP<const MV> >(X1));
          }
          TEUCHOS_TEST_FOR_EXCEPTION(initialX2Rank != sizeX2,
                             std::runtime_error,
                             "projectAndNormalize(X2,X1) returned rank "
                             << initialX2Rank << " from " << sizeX2
                             << " vectors. Cannot continue.");
          debugOut << "done." << endl
                   << "Calling orthonormError() on X2... ";
          err = OM->orthonormError (*X2);
          TEUCHOS_TEST_FOR_EXCEPTION(err > TOL,
                             std::runtime_error,
                             "projectAndNormalize(X2,X1) did not meet tolerance: "
                             "orthonormError(X2) = " << err << " > TOL = " << TOL);
          debugOut << "done: || <X2,X2> - I || = " << err << endl
                   << "Calling orthogError(X2, X1)... ";
          err = OM->orthogError (*X2, *X1);
          TEUCHOS_TEST_FOR_EXCEPTION(err > TOL,
                             std::runtime_error,
                             "projectAndNormalize(X2,X1) did not meet tolerance: "
                             "orthogError(X2,X1) = " << err << " > TOL = " << TOL);
          debugOut << "done: || <X2,X1> || = " << err << endl;
        }


        //
        // If OM is an OutOfPlaceNormalizerMixin, exercise the
        // out-of-place normalization routines.
        //
        typedef Belos::OutOfPlaceNormalizerMixin<Scalar, MV> mixin_type;
        RCP<mixin_type> tsqr = rcp_dynamic_cast<mixin_type>(OM);
        if (! tsqr.is_null())
          {
            magnitude_type err;
            debugOut << endl
                     << "=== OutOfPlaceNormalizerMixin tests ==="
                     << endl << endl;
            //
            // Fill X1_in with random values, and test the normalization
            // error with normalizeOutOfPlace().
            //
            // Don't overwrite X1, else you'll mess up the tests that
            // follow!
            //
            RCP<MV> X1_in = MVT::CloneCopy (*X1);
            debugOut << "Filling X1_in with random values... ";
            MVT::MvRandom(*X1_in);
            debugOut << "done." << endl;
            debugOut << "Filling X1_out with different random values...";
            RCP<MV> X1_out = MVT::Clone(*X1_in, MVT::GetNumberVecs(*X1_in));
            MVT::MvRandom(*X1_out);
            debugOut << "done." << endl
                     << "Calling normalizeOutOfPlace(*X1_in, *X1_out, null)... ";
            const int initialX1Rank =
              tsqr->normalizeOutOfPlace(*X1_in, *X1_out, Teuchos::null);
            TEUCHOS_TEST_FOR_EXCEPTION(initialX1Rank != sizeX1, std::runtime_error,
                               "normalizeOutOfPlace(*X1_in, *X1_out, null) "
                               "returned rank " << initialX1Rank << " from "
                               << sizeX1 << " vectors. Cannot continue.");
            debugOut << "done." << endl
                     << "Calling orthonormError() on X1_out... ";
            err = OM->orthonormError(*X1_out);
            TEUCHOS_TEST_FOR_EXCEPTION(err > TOL, std::runtime_error,
                               "After calling normalizeOutOfPlace(*X1_in, "
                               "*X1_out, null), orthonormError(X1) = "
                               << err << " > TOL = " << TOL);
            debugOut << "done: ||<X1_out,X1_out> - I|| = " << err << endl;

            //
            // Fill X2_in with random values, project against X1_out
            // and normalize via projectAndNormalizeOutOfPlace(), and
            // test the orthogonalization error.
            //
            // Don't overwrite X2, else you'll mess up the tests that
            // follow!
            //
            RCP<MV> X2_in = MVT::CloneCopy (*X2);
            debugOut << "Filling X2_in with random values... ";
            MVT::MvRandom(*X2_in);
            debugOut << "done." << endl
                     << "Filling X2_out with different random values...";
            RCP<MV> X2_out = MVT::Clone(*X2_in, MVT::GetNumberVecs(*X2_in));
            MVT::MvRandom(*X2_out);
            debugOut << "done." << endl
                     << "Calling projectAndNormalizeOutOfPlace(X2_in, X2_out, "
                     << "C, B, X1_out)...";
            int initialX2Rank;
            {
              Array<RCP<mat_type> > C (1);
              RCP<mat_type> B = Teuchos::null;
              initialX2Rank =
                tsqr->projectAndNormalizeOutOfPlace (*X2_in, *X2_out, C, B,
                                                     tuple<RCP<const MV> >(X1_out));
            }
            TEUCHOS_TEST_FOR_EXCEPTION(initialX2Rank != sizeX2,
                               std::runtime_error,
                               "projectAndNormalizeOutOfPlace(*X2_in, "
                               "*X2_out, C, B, tuple(X1_out)) returned rank "
                               << initialX2Rank << " from " << sizeX2
                               << " vectors. Cannot continue.");
            debugOut << "done." << endl
                     << "Calling orthonormError() on X2_out... ";
            err = OM->orthonormError (*X2_out);
            TEUCHOS_TEST_FOR_EXCEPTION(err > TOL, std::runtime_error,
                               "projectAndNormalizeOutOfPlace(*X2_in, *X2_out, "
                               "C, B, tuple(X1_out)) did not meet tolerance: "
                               "orthonormError(X2_out) = "
                               << err << " > TOL = " << TOL);
            debugOut << "done: || <X2_out,X2_out> - I || = " << err << endl
                     << "Calling orthogError(X2_out, X1_out)... ";
            err = OM->orthogError (*X2_out, *X1_out);
            TEUCHOS_TEST_FOR_EXCEPTION(err > TOL, std::runtime_error,
                               "projectAndNormalizeOutOfPlace(*X2_in, *X2_out, "
                               "C, B, tuple(X1_out)) did not meet tolerance: "
                               "orthogError(X2_out, X1_out) = "
                               << err << " > TOL = " << TOL);
            debugOut << "done: || <X2_out,X1_out> || = " << err << endl;
            debugOut << endl
                     << "=== Done with OutOfPlaceNormalizerMixin tests ==="
                     << endl << endl;
          }

        {
          //
          // Test project() on a random multivector S, by projecting S
          // against various combinations of X1 and X2.
          //
          MVT::MvRandom(*S);

          debugOut << "Testing project() by projecting a random multivector S "
            "against various combinations of X1 and X2 " << endl;
          const int thisNumFailed = testProject(OM,S,X1,X2,MyOM);
          numFailed += thisNumFailed;
          if (thisNumFailed > 0)
            debugOut << "  *** " << thisNumFailed
                     << (thisNumFailed > 1 ? " tests" : " test")
                     << " failed." << endl;
        }

        if (isRankRevealing)
          {
            // run a X1,Y2 range multivector against P_{X1,X1} P_{Y2,Y2}
            // note, this is allowed under the restrictions on project(),
            // because <X1,Y2> = 0
            // also, <Y2,Y2> = I, but <X1,X1> != I, so biOrtho must be set to false
            // it should require randomization, as
            // P_{X1,X1} P_{Y2,Y2} (X1*C1 + Y2*C2) = P_{X1,X1} X1*C1 = 0
            mat_type C1(sizeX1,sizeS), C2(sizeX2,sizeS);
            C1.random();
            C2.random();
            // S := X1*C1
            MVT::MvTimesMatAddMv(ONE,*X1,C1,ZERO,*S);
            // S := S + X2*C2
            MVT::MvTimesMatAddMv(ONE,*X2,C2,ONE,*S);

            debugOut << "Testing project() by projecting [X1 X2]-range multivector "
              "against P_X1 P_X2 " << endl;
            const int thisNumFailed = testProject(OM,S,X1,X2,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }

        // This test is only distinct from the rank-1 multivector test
        // (below) if S has at least 3 columns.
        if (isRankRevealing && sizeS > 2)
          {
            MVT::MvRandom(*S);
            RCP<MV> mid = MVT::Clone(*S,1);
            mat_type c(sizeS,1);
            MVT::MvTimesMatAddMv(ONE,*S,c,ZERO,*mid);
            std::vector<int> ind(1);
            ind[0] = sizeS-1;
            MVT::SetBlock(*mid,ind,*S);

            debugOut << "Testing normalize() on a rank-deficient multivector " << endl;
            const int thisNumFailed = testNormalize(OM,S,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }

        // This test will only exercise rank deficiency if S has at least 2
        // columns.
        if (isRankRevealing && sizeS > 1)
          {
            // rank-1
            RCP<MV> one = MVT::Clone(*S,1);
            MVT::MvRandom(*one);
            // put multiple of column 0 in columns 0:sizeS-1
            for (int i=0; i<sizeS; i++)
              {
                std::vector<int> ind(1);
                ind[0] = i;
                RCP<MV> Si = MVT::CloneViewNonConst(*S,ind);
                MVT::MvAddMv(SCT::random(),*one,ZERO,*one,*Si);
              }
            debugOut << "Testing normalize() on a rank-1 multivector " << endl;
            const int thisNumFailed = testNormalize(OM,S,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }

        {
          std::vector<int> ind(1);
          MVT::MvRandom(*S);

          debugOut << "Testing projectAndNormalize() on a random multivector " << endl;
          const int thisNumFailed = testProjectAndNormalize(OM,S,X1,X2,MyOM);
          numFailed += thisNumFailed;
          if (thisNumFailed > 0)
            debugOut << "  *** " << thisNumFailed
                     << (thisNumFailed > 1 ? " tests" : " test")
                     << " failed." << endl;
        }

        if (isRankRevealing)
          {
            // run a X1,X2 range multivector against P_X1 P_X2
            // this is allowed as <X1,X2> == 0
            // it should require randomization, as
            // P_X1 P_X2 (X1*C1 + X2*C2) = P_X1 X1*C1 = 0
            // and
            // P_X2 P_X1 (X2*C2 + X1*C1) = P_X2 X2*C2 = 0
            mat_type C1(sizeX1,sizeS), C2(sizeX2,sizeS);
            C1.random();
            C2.random();
            MVT::MvTimesMatAddMv(ONE,*X1,C1,ZERO,*S);
            MVT::MvTimesMatAddMv(ONE,*X2,C2,ONE,*S);

            debugOut << "Testing projectAndNormalize() by projecting [X1 X2]-range "
              "multivector against P_X1 P_X2 " << endl;
            const int thisNumFailed = testProjectAndNormalize(OM,S,X1,X2,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }

        // This test is only distinct from the rank-1 multivector test
        // (below) if S has at least 3 columns.
        if (isRankRevealing && sizeS > 2)
          {
            MVT::MvRandom(*S);
            RCP<MV> mid = MVT::Clone(*S,1);
            mat_type c(sizeS,1);
            MVT::MvTimesMatAddMv(ONE,*S,c,ZERO,*mid);
            std::vector<int> ind(1);
            ind[0] = sizeS-1;
            MVT::SetBlock(*mid,ind,*S);

            debugOut << "Testing projectAndNormalize() on a rank-deficient "
              "multivector " << endl;
            const int thisNumFailed = testProjectAndNormalize(OM,S,X1,X2,MyOM);
            numFailed += thisNumFailed;
            if (thisNumFailed > 0)
              debugOut << "  *** " << thisNumFailed
                       << (thisNumFailed > 1 ? " tests" : " test")
                       << " failed." << endl;
          }

        // This test will only exercise rank deficiency if S has at least 2
        // columns.
        if (isRankRevealing && sizeS > 1)
          {
            // rank-1
            RCP<MV> one = MVT::Clone(*S,1);
            MVT::MvRandom(*one);
            // Put a multiple of column 0 in columns 0:sizeS-1.
            for (int i=0; i<sizeS; i++)
              {
                std::vector<int> ind(1);
                ind[0] = i;
                RCP<MV> Si = MVT::CloneViewNonConst(*S,ind);
                MVT::MvAddMv(SCT::random(),*one,ZERO,*one,*Si);
              }
            debugOut << "Testing projectAndNormalize() on a rank-1 multivector " << endl;
            bool constantStride = true;
            if (! MVT::HasConstantStride(*S)) {
              debugOut << "-- S does not have constant stride" << endl;
              constantStride = false;
            }
            if (! MVT::HasConstantStride(*X1)) {
              debugOut << "-- X1 does not have constant stride" << endl;
              constantStride = false;
            }
            if (! MVT::HasConstantStride(*X2)) {
              debugOut << "-- X2 does not have constant stride" << endl;
              constantStride = false;
            }
            if (! constantStride) {
              debugOut << "-- Skipping this test, since TSQR does not work on "
                "multivectors with nonconstant stride" << endl;
            }
            else {
              const int thisNumFailed = testProjectAndNormalize(OM,S,X1,X2,MyOM);
              numFailed += thisNumFailed;
              if (thisNumFailed > 0) {
                debugOut << "  *** " << thisNumFailed
                         << (thisNumFailed > 1 ? " tests" : " test")
                         << " failed." << endl;
              }
            }
          }

        if (numFailed != 0) {
          MyOM->stream(Errors) << numFailed << " total test failures." << endl;
        }
        return numFailed;
      }

    private:

      /// \fn MVDiff
      ///
      /// Compute and return $\|X - Y\|_F$, the Frobenius (sum of
      /// squares) norm of the difference between X and Y.
      static magnitude_type
      MVDiff (const MV& X, const MV& Y)
      {
        using Teuchos::RCP;

        const scalar_type ONE = SCT::one();
        const int numCols = MVT::GetNumberVecs(X);
        TEUCHOS_TEST_FOR_EXCEPTION( (MVT::GetNumberVecs(Y) != numCols),
                            std::logic_error,
                            "MVDiff: X and Y should have the same number of columns."
                            "  X has " << numCols << " column(s) and Y has "
                            << MVT::GetNumberVecs(Y) << " columns." );
        // Resid := X
        RCP< MV > Resid = MVT::CloneCopy(X);
        // Resid := Resid - Y
        MVT::MvAddMv (-ONE, Y, ONE, *Resid, *Resid);

        return frobeniusNorm (*Resid);
      }


      /// \fn frobeniusNorm
      ///
      /// Compute and return the Frobenius norm of X.
      static magnitude_type
      frobeniusNorm (const MV& X)
      {
        const scalar_type ONE = SCT::one();
        const int numCols = MVT::GetNumberVecs(X);
        mat_type C (numCols, numCols);

        // $C := X^* X$
        MVT::MvTransMv (ONE, X, X, C);

        magnitude_type err (0);
        for (int i = 0; i < numCols; ++i)
          err += SCT::magnitude (C(i,i));

        return SCT::magnitude (SCT::squareroot (err));
      }


      static int
      testProjectAndNormalize (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                               const Teuchos::RCP< const MV >& S,
                               const Teuchos::RCP< const MV >& X1,
                               const Teuchos::RCP< const MV >& X2,
                               const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        return testProjectAndNormalizeNew (OM, S, X1, X2, MyOM);
      }

      /// Test OrthoManager::projectAndNormalize() for the specific
      /// OrthoManager instance.
      ///
      /// \return Count of errors (should be zero)
      static int
      testProjectAndNormalizeOld (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > >& OM,
                                  const Teuchos::RCP< const MV >& S,
                                  const Teuchos::RCP< const MV >& X1,
                                  const Teuchos::RCP< const MV >& X2,
                                  const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;

        const scalar_type ONE = SCT::one();
        const magnitude_type ZERO = SCT::magnitude(SCT::zero());

        // Relative tolerance against which all tests are performed.
        const magnitude_type TOL = 1.0e-12;
        // Absolute tolerance constant
        const magnitude_type ATOL = 10;

        const int sizeS = MVT::GetNumberVecs(*S);
        const int sizeX1 = MVT::GetNumberVecs(*X1);
        const int sizeX2 = MVT::GetNumberVecs(*X2);
        int numerr = 0;
        std::ostringstream sout;

        //
        // output tests:
        //   <S_out,S_out> = I
        //   <S_out,X1> = 0
        //   <S_out,X2> = 0
        //   S_in = S_out B + X1 C1 + X2 C2
        //
        // we will loop over an integer specifying the test combinations
        // the bit pattern for the different tests is listed in parenthesis
        //
        // for the projectors, test the following combinations:
        // none              (00)
        // P_X1              (01)
        // P_X2              (10)
        // P_X1 P_X2         (11)
        // P_X2 P_X1         (11)
        // the latter two should be tested to give the same answer
        //
        // for each of these, we should test with C1, C2 and B
        //
        // if hasM:
        // with and without MX1   (1--)
        // with and without MX2  (1---)
        // with and without MS  (1----)
        //
        // as hasM controls the upper level bits, we need only run test cases 0-3 if hasM==false
        // otherwise, we run test cases 0-31
        //

        int numtests = 4;

        // test ortho error before orthonormalizing
        if (X1 != null) {
          magnitude_type err = OM->orthogError(*S,*X1);
          sout << "   || <S,X1> || before     : " << err << endl;
        }
        if (X2 != null) {
          magnitude_type err = OM->orthogError(*S,*X2);
          sout << "   || <S,X2> || before     : " << err << endl;
        }

        for (int t=0; t<numtests; t++) {

          Array< RCP< const MV > > theX;
          RCP<mat_type > B = rcp( new mat_type(sizeS,sizeS) );
          Array<RCP<mat_type > > C;
          if ( (t % 3) == 0 ) {
            // neither <X1,Y1> nor <X2,Y2>
            // C, theX and theY are already empty
          }
          else if ( (t % 3) == 1 ) {
            // X1
            theX = tuple(X1);
            C = tuple( rcp(new mat_type(sizeX1,sizeS)) );
          }
          else if ( (t % 3) == 2 ) {
            // X2
            theX = tuple(X2);
            C = tuple( rcp(new mat_type(sizeX2,sizeS)) );
          }
          else {
            // X1 and X2, and the reverse.
            theX = tuple(X1,X2);
            C = tuple( rcp(new mat_type(sizeX1,sizeS)),
                       rcp(new mat_type(sizeX2,sizeS)) );
          }

          // We wrap up all the OrthoManager calls in a try-catch
          // block, in order to check whether any of the methods throw
          // an exception.  For the tests we perform, every thrown
          // exception is a failure.
          try {
            // call routine
            // if (t && 3) == 3, {
            //    call with reversed input: X2 X1
            // }
            // test all outputs for correctness
            // test all outputs for equivalence

            // here is where the outputs go
            Array<RCP<MV> > S_outs;
            Array<Array<RCP<mat_type > > > C_outs;
            Array<RCP<mat_type > > B_outs;
            RCP<MV> Scopy;
            Array<int> ret_out;

            // copies of S,MS
            Scopy = MVT::CloneCopy(*S);
            // randomize this data, it should be overwritten
            B->random();
            for (size_type i=0; i<C.size(); i++) {
              C[i]->random();
            }
            // Run test.  Since S was specified by the caller and
            // Scopy is a copy of S, we don't know what rank to expect
            // here -- though we do require that S have rank at least
            // one.
            //
            // Note that Anasazi and Belos differ, among other places,
            // in the order of arguments to projectAndNormalize().
            int ret = OM->projectAndNormalize(*Scopy,C,B,theX);
            sout << "projectAndNormalize() returned rank " << ret << endl;
            if (ret == 0) {
              sout << "  *** Error: returned rank is zero, cannot continue tests" << endl;
              numerr++;
              break;
            }
            ret_out.push_back(ret);
            // projectAndNormalize() is only required to return a
            // basis of rank "ret"
            // this is what we will test:
            //   the first "ret" columns in Scopy
            //   the first "ret" rows in B
            // save just the parts that we want
            // we allocate S and MS for each test, so we can save these as views
            // however, save copies of the C and B
            if (ret < sizeS) {
              std::vector<int> ind(ret);
              for (int i=0; i<ret; i++) {
                ind[i] = i;
              }
              S_outs.push_back( MVT::CloneViewNonConst(*Scopy,ind) );
              B_outs.push_back( rcp( new mat_type(Teuchos::Copy,*B,ret,sizeS) ) );
            }
            else {
              S_outs.push_back( Scopy );
              B_outs.push_back( rcp( new mat_type(*B) ) );
            }
            C_outs.push_back( Array<RCP<mat_type > >(0) );
            if (C.size() > 0) {
              C_outs.back().push_back( rcp( new mat_type(*C[0]) ) );
            }
            if (C.size() > 1) {
              C_outs.back().push_back( rcp( new mat_type(*C[1]) ) );
            }

            // do we run the reversed input?
            if ( (t % 3) == 3 ) {
              // copies of S,MS
              Scopy = MVT::CloneCopy(*S);

              // Fill the B and C[i] matrices with random data.  The
              // data will be overwritten by projectAndNormalize().
              // Filling these matrices here is only to catch some
              // bugs in projectAndNormalize().
              B->random();
              for (size_type i=0; i<C.size(); i++) {
                C[i]->random();
              }
              // flip the inputs
              theX = tuple( theX[1], theX[0] );
              // Run test.
              // Note that Anasazi and Belos differ, among other places,
              // in the order of arguments to projectAndNormalize().
              ret = OM->projectAndNormalize(*Scopy,C,B,theX);
              sout << "projectAndNormalize() returned rank " << ret << endl;
              if (ret == 0) {
                sout << "  *** Error: returned rank is zero, cannot continue tests" << endl;
                numerr++;
                break;
              }
              ret_out.push_back(ret);
              // projectAndNormalize() is only required to return a
              // basis of rank "ret"
              // this is what we will test:
              //   the first "ret" columns in Scopy
              //   the first "ret" rows in B
              // save just the parts that we want
              // we allocate S and MS for each test, so we can save these as views
              // however, save copies of the C and B
              if (ret < sizeS) {
                std::vector<int> ind(ret);
                for (int i=0; i<ret; i++) {
                  ind[i] = i;
                }
                S_outs.push_back( MVT::CloneViewNonConst(*Scopy,ind) );
                B_outs.push_back( rcp( new mat_type(Teuchos::Copy,*B,ret,sizeS) ) );
              }
              else {
                S_outs.push_back( Scopy );
                B_outs.push_back( rcp( new mat_type(*B) ) );
              }
              C_outs.push_back( Array<RCP<mat_type > >() );
              // reverse the Cs to compensate for the reverse projectors
              C_outs.back().push_back( rcp( new mat_type(*C[1]) ) );
              C_outs.back().push_back( rcp( new mat_type(*C[0]) ) );
              // flip the inputs back
              theX = tuple( theX[1], theX[0] );
            }


            // test all outputs for correctness
            for (size_type o=0; o<S_outs.size(); o++) {
              // S^T M S == I
              {
                magnitude_type err = OM->orthonormError(*S_outs[o]);
                if (err > TOL) {
                  sout << endl
                       << "  *** Test (number " << (t+1) << " of " << numtests
                       << " total tests) failed: Tolerance exceeded!  Error = "
                       << err << " > TOL = " << TOL << "."
                       << endl << endl;
                  numerr++;
                }
                sout << "   || <S,S> - I || after  : " << err << endl;
              }
              // S_in = X1*C1 + C2*C2 + S_out*B
              {
                RCP<MV> tmp = MVT::Clone(*S,sizeS);
                MVT::MvTimesMatAddMv(ONE,*S_outs[o],*B_outs[o],ZERO,*tmp);
                if (C_outs[o].size() > 0) {
                  MVT::MvTimesMatAddMv(ONE,*X1,*C_outs[o][0],ONE,*tmp);
                  if (C_outs[o].size() > 1) {
                    MVT::MvTimesMatAddMv(ONE,*X2,*C_outs[o][1],ONE,*tmp);
                  }
                }
                magnitude_type err = MVDiff(*tmp,*S);
                if (err > ATOL*TOL) {
                  sout << endl
                       << "  *** Test (number " << (t+1) << " of " << numtests
                       << " total tests) failed: Tolerance exceeded!  Error = "
                       << err << " > ATOL*TOL = " << (ATOL*TOL) << "."
                       << endl << endl;
                  numerr++;
                }
                sout << "  " << t << "|| S_in - X1*C1 - X2*C2 - S_out*B || : " << err << endl;
              }
              // <X1,S> == 0
              if (theX.size() > 0 && theX[0] != null) {
                magnitude_type err = OM->orthogError(*theX[0],*S_outs[o]);
                if (err > TOL) {
                  sout << endl
                       << "  *** Test (number " << (t+1) << " of " << numtests
                       << " total tests) failed: Tolerance exceeded!  Error = "
                       << err << " > TOL = " << TOL << "."
                       << endl << endl;
                  numerr++;
                }
                sout << "  " << t << "|| <X[0],S> || after      : " << err << endl;
              }
              // <X2,S> == 0
              if (theX.size() > 1 && theX[1] != null) {
                magnitude_type err = OM->orthogError(*theX[1],*S_outs[o]);
                if (err > TOL) {
                  sout << endl
                       << "  *** Test (number " << (t+1) << " of " << numtests
                       << " total tests) failed: Tolerance exceeded!  Error = "
                       << err << " > TOL = " << TOL << "."
                       << endl << endl;
                  numerr++;
                }
                sout << "  " << t << "|| <X[1],S> || after      : " << err << endl;
              }
            }
          }
          catch (Belos::OrthoError& e) {
            sout << "  *** Error: OrthoManager threw exception: " << e.what() << endl;
            numerr++;
          }

        } // test for

        // NOTE (mfh 05 Nov 2010) Since Belos::MsgType is an enum,
        // doing bitwise logical computations on Belos::MsgType values
        // (such as "Debug | Errors") and passing the result into
        // MyOM->stream() confuses the compiler.  As a result, we have
        // to do some type casts to make it work.
        const int msgType = (numerr > 0) ?
          (static_cast<int>(Debug) | static_cast<int>(Errors)) :
          static_cast<int>(Debug);

        // We report debug-level messages always.  We also report
        // errors if at least one test failed.
        MyOM->stream(static_cast< MsgType >(msgType)) << sout.str() << endl;
        return numerr;
      }

      /// Test OrthoManager::normalize() for the specific OrthoManager
      /// instance.
      ///
      /// \return Count of errors (should be zero)
      static int
      testNormalize (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > >& OM,
                     const Teuchos::RCP< const MV >& S,
                     const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;

        const scalar_type ONE = SCT::one();
        std::ostringstream sout;
        // Total number of failed tests in this call of this routine.
        int numerr = 0;

        // Relative tolerance against which all tests are performed.
        // We are measuring things in the Frobenius norm $\| \cdot \|_F$.
        // The following bounds hold for all $m \times n$ matrices $A$:
        // \[
        // \|A\|_2 \leq \|A\|_F \leq \sqrt{r} \|A\|_2,
        // \]
        // where $r$ is the (column) rank of $A$.  We bound this above
        // by the number of columns in $A$.
        //
        // An accurate normalization in the Euclidean norm of a matrix
        // $A$ with at least as many rows m as columns n, should
        // produce orthogonality $\|Q^* Q - I\|_2$ less than a factor
        // of machine precision times a low-order polynomial in m and
        // n, and residual $\|A - Q B\|_2$ (where $A = Q B$ is the
        // computed normalization) less than that bound times the norm
        // of $A$.
        //
        // Since we are measuring both of these quantitites in the
        // Frobenius norm instead, we should scale this bound by
        // $\sqrt{n}$.

        const int numRows = MVT::GetGlobalLength(*S);
        const int numCols = MVT::GetNumberVecs(*S);
        const int sizeS = MVT::GetNumberVecs(*S);

        // A good heuristic is to scale the bound by the square root
        // of the number of floating-point operations.  One could
        // perhaps support this theoretically, since we are using
        // uniform random test problems.
        const magnitude_type fudgeFactor =
          SMT::squareroot(magnitude_type(numRows) *
                          magnitude_type(numCols) *
                          magnitude_type(numCols));
        const magnitude_type TOL = SMT::eps() * fudgeFactor *
          SMT::squareroot(magnitude_type(numCols));

        // Absolute tolerance scaling: the Frobenius norm of the test
        // matrix S.  TOL*ATOL is the absolute tolerance for the
        // residual $\|A - Q*B\|_F$.
        const magnitude_type ATOL = frobeniusNorm (*S);

        sout << "The test matrix S has Frobenius norm " << ATOL
             << ", and the relative error tolerance is TOL = "
             << TOL << "." << endl;

        const int numtests = 1;
        for (int t = 0; t < numtests; ++t) {

          try {
            // call routine
            // test all outputs for correctness

            // S_copy gets a copy of S; we normalize in place, so we
            // need a copy to check whether the normalization
            // succeeded.
            RCP< MV > S_copy = MVT::CloneCopy (*S);

            // Matrix of coefficients from the normalization.
            RCP< mat_type > B (new mat_type (sizeS, sizeS));
            // The contents of B will be overwritten, but fill with
            // random data just to make sure that the normalization
            // operated on all the elements of B on which it should
            // operate.
            B->random();

            const int reportedRank = OM->normalize (*S_copy, B);
            sout << "normalize() returned rank " << reportedRank << endl;
            if (reportedRank == 0) {
              sout << "  *** Error: Cannot continue, since normalize() "
                "reports that S has rank 0" << endl;
              numerr++;
              break;
            }
            //
            // We don't know in this routine whether the input
            // multivector S has full rank; it is only required to
            // have nonzero rank.  Thus, we extract the first
            // reportedRank columns of S_copy and the first
            // reportedRank rows of B, and perform tests on them.
            //

            // Construct S_view, a view of the first reportedRank
            // columns of S_copy.
            std::vector<int> indices (reportedRank);
            for (int j = 0; j < reportedRank; ++j)
              indices[j] = j;
            RCP< MV > S_view = MVT::CloneViewNonConst (*S_copy, indices);
            // Construct B_top, a copy of the first reportedRank rows
            // of B.
            //
            // NOTE: We create this as a copy and not a view, because
            // otherwise it would not be safe with respect to RCPs.
            // This is because mat_type uses raw pointers
            // inside, so that a view would become invalid when B
            // would fall out of scope.
            RCP< mat_type > B_top (new mat_type (Teuchos::Copy, *B, reportedRank, sizeS));

            // Check ||<S_view,S_view> - I||
            {
              const magnitude_type err = OM->orthonormError(*S_view);
              if (err > TOL) {
                sout << "  *** Error: Tolerance exceeded: err = "
                     << err << " > TOL = " << TOL << endl;
                numerr++;
              }
              sout << "   || <S,S> - I || after  : " << err << endl;
            }
            // Check the residual ||Residual|| = ||S_view * B_top -
            // S_orig||, where S_orig is a view of the first
            // reportedRank columns of S.
            {
              // Residual is allocated with reportedRank columns.  It
              // will contain the result of testing the residual error
              // of the normalization (i.e., $\|S - S_in*B\|$).  It
              // should have the dimensions of S.  Its initial value
              // is a copy of the first reportedRank columns of S.
              RCP< MV > Residual = MVT::CloneCopy (*S);

              // Residual := Residual - S_view * B_view
              MVT::MvTimesMatAddMv (-ONE, *S_view, *B_top, ONE, *Residual);

              // Compute ||Residual||
              const magnitude_type err = frobeniusNorm (*Residual);
              if (err > ATOL*TOL) {
                sout << "  *** Error: Tolerance exceeded: err = "
                     << err << " > ATOL*TOL = " << (ATOL*TOL) << endl;
                numerr++;
              }
              sout << "  " << t << "|| S - Q*B || : " << err << endl;
            }
          }
          catch (Belos::OrthoError& e) {
            sout << "  *** Error: the OrthoManager's normalize() method "
              "threw an exception: " << e.what() << endl;
            numerr++;
          }

        } // test for

        const MsgType type = (numerr == 0) ? Debug : static_cast<MsgType> (static_cast<int>(Errors) | static_cast<int>(Debug));
        MyOM->stream(type) << sout.str();
        MyOM->stream(type) << endl;

        return numerr;
      }

      /// Test OrthoManager::projectAndNormalize() for the specific
      /// OrthoManager instance.
      ///
      /// \return Count of errors (should be zero)
      static int
      testProjectAndNormalizeNew (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                                  const Teuchos::RCP< const MV >& S,
                                  const Teuchos::RCP< const MV >& X1,
                                  const Teuchos::RCP< const MV >& X2,
                                  const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;

        // We collect all the output in this string wrapper, and print
        // it at the end.
        std::ostringstream sout;
        // Total number of failed tests in this call of this routine.
        int numerr = 0;

        const int numRows = MVT::GetGlobalLength(*S);
        const int numCols = MVT::GetNumberVecs(*S);
        const int sizeS = MVT::GetNumberVecs(*S);

        // Relative tolerance against which all tests are performed.
        // We are measuring things in the Frobenius norm $\| \cdot \|_F$.
        // The following bounds hold for all $m \times n$ matrices $A$:
        // \[
        // \|A\|_2 \leq \|A\|_F \leq \sqrt{r} \|A\|_2,
        // \]
        // where $r$ is the (column) rank of $A$.  We bound this above
        // by the number of columns in $A$.
        //
        // Since we are measuring both of these quantitites in the
        // Frobenius norm instead, we scale all error tests by
        // $\sqrt{n}$.
        //
        // A good heuristic is to scale the bound by the square root
        // of the number of floating-point operations.  One could
        // perhaps support this theoretically, since we are using
        // uniform random test problems.
        const magnitude_type fudgeFactor =
          SMT::squareroot(magnitude_type(numRows) *
                          magnitude_type(numCols) *
                          magnitude_type(numCols));
        const magnitude_type TOL = SMT::eps() * fudgeFactor *
          SMT::squareroot(magnitude_type(numCols));

        // Absolute tolerance scaling: the Frobenius norm of the test
        // matrix S.  TOL*ATOL is the absolute tolerance for the
        // residual $\|A - Q*B\|_F$.
        const magnitude_type ATOL = frobeniusNorm (*S);

        sout << "-- The test matrix S has Frobenius norm " << ATOL
             << ", and the relative error tolerance is TOL = "
             << TOL << "." << endl;

        // Q will contain the result of projectAndNormalize() on S.
        RCP< MV > Q = MVT::CloneCopy(*S);
        // We use this for collecting the residual error components
        RCP< MV > Residual = MVT::CloneCopy(*S);
        // Number of elements in the X array of blocks against which
        // to project S.
        const int num_X = 2;
        Array< RCP< const MV > > X (num_X);
        X[0] = MVT::CloneCopy(*X1);
        X[1] = MVT::CloneCopy(*X2);

        // Coefficients for the normalization
        RCP< mat_type > B (new mat_type (sizeS, sizeS));

        // Array of coefficients matrices from the projection.
        // For our first test, we allocate each of these matrices
        // with the proper dimensions.
        Array< RCP< mat_type > > C (num_X);
        for (int k = 0; k < num_X; ++k)
          {
            C[k] = rcp (new mat_type (MVT::GetNumberVecs(*X[k]), sizeS));
            C[k]->random(); // will be overwritten
          }
        try {
          // Q*B := (I - X X^*) S
          const int reportedRank = OM->projectAndNormalize (*Q, C, B, X);

          // Pick out the first reportedRank columns of Q.
          std::vector<int> indices (reportedRank);
          for (int j = 0; j < reportedRank; ++j)
            indices[j] = j;
          RCP< const MV > Q_left = MVT::CloneView (*Q, indices);

          // Test whether the first reportedRank columns of Q are
          // orthogonal.
          {
            const magnitude_type orthoError = OM->orthonormError (*Q_left);
            sout << "-- ||Q(1:" << reportedRank << ")^* Q(1:" << reportedRank
                 << ") - I||_F = " << orthoError << endl;
            if (orthoError > TOL)
              {
                sout << "   *** Error: ||Q(1:" << reportedRank << ")^* Q(1:"
                     << reportedRank << ") - I||_F = " << orthoError
                     << " > TOL = " << TOL << "." << endl;
                numerr++;
              }
          }

          // Compute the residual: if successful, S = Q*B +
          // X (X^* S =: C) in exact arithmetic.  So, the residual is
          // S - Q*B - X1 C1 - X2 C2.
          //
          // Residual := S
          MVT::MvAddMv (SCT::one(), *S, SCT::zero(), *Residual, *Residual);
          {
            // Pick out the first reportedRank rows of B.  Make a deep
            // copy, since mat_type is not safe with respect
            // to RCP-based memory management (it uses raw pointers
            // inside).
            RCP< const mat_type > B_top (new mat_type (Teuchos::Copy, *B, reportedRank, B->numCols()));
            // Residual := Residual - Q(:, 1:reportedRank) * B(1:reportedRank, :)
            MVT::MvTimesMatAddMv (-SCT::one(), *Q_left, *B_top, SCT::one(), *Residual);
          }
          // Residual := Residual - X[k]*C[k]
          for (int k = 0; k < num_X; ++k)
            MVT::MvTimesMatAddMv (-SCT::one(), *X[k], *C[k], SCT::one(), *Residual);
          const magnitude_type residErr = frobeniusNorm (*Residual);
          sout << "-- ||S - Q(:, 1:" << reportedRank << ")*B(1:"
               << reportedRank << ", :) - X1*C1 - X2*C2||_F = "
               << residErr << endl;
          if (residErr > ATOL * TOL)
            {
              sout << "   *** Error: ||S - Q(:, 1:" << reportedRank
                   << ")*B(1:" << reportedRank << ", :) "
                   << "- X1*C1 - X2*C2||_F = " << residErr
                   << " > ATOL*TOL = " << (ATOL*TOL) << "." << endl;
              numerr++;
            }
          // Verify that Q(1:reportedRank) is orthogonal to X[k], for
          // all k.  This test only makes sense if reportedRank > 0.
          if (reportedRank == 0)
            {
              sout << "-- Reported rank of Q is zero: skipping Q, X[k] "
                "orthogonality test." << endl;
            }
          else
            {
              for (int k = 0; k < num_X; ++k)
                {
                  // Q should be orthogonal to X[k], for all k.
                  const magnitude_type projErr = OM->orthogError(*X[k], *Q_left);
                  sout << "-- ||<Q(1:" << reportedRank << "), X[" << k
                       << "]>||_F = " << projErr << endl;
                  if (projErr > ATOL*TOL)
                    {
                      sout << "   *** Error: ||<Q(1:" << reportedRank << "), X["
                           << k << "]>||_F = " << projErr << " > ATOL*TOL = "
                           << (ATOL*TOL) << "." << endl;
                      numerr++;
                    }
                }
            }
        } catch (Belos::OrthoError& e) {
          sout << "  *** Error: The OrthoManager subclass instance threw "
            "an exception: " << e.what() << endl;
          numerr++;
        }

        // Print out the collected diagnostic messages, which possibly
        // include error messages.
        const MsgType type = (numerr == 0) ? Debug : static_cast<MsgType> (static_cast<int>(Errors) | static_cast<int>(Debug));
        MyOM->stream(type) << sout.str();
        MyOM->stream(type) << endl;

        return numerr;
      }


      /// Test OrthoManager::project() for the specific OrthoManager instance.
      ///
      /// \return Count of errors (should be zero)
      static int
      testProjectNew (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                      const Teuchos::RCP< const MV >& S,
                      const Teuchos::RCP< const MV >& X1,
                      const Teuchos::RCP< const MV >& X2,
                      const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;

        // We collect all the output in this string wrapper, and print
        // it at the end.
        std::ostringstream sout;
        // Total number of failed tests in this call of this routine.
        int numerr = 0;

        const int numRows = MVT::GetGlobalLength(*S);
        const int numCols = MVT::GetNumberVecs(*S);
        const int sizeS = MVT::GetNumberVecs(*S);

        // Relative tolerance against which all tests are performed.
        // We are measuring things in the Frobenius norm $\| \cdot \|_F$.
        // The following bounds hold for all $m \times n$ matrices $A$:
        // \[
        // \|A\|_2 \leq \|A\|_F \leq \sqrt{r} \|A\|_2,
        // \]
        // where $r$ is the (column) rank of $A$.  We bound this above
        // by the number of columns in $A$.
        //
        // Since we are measuring both of these quantitites in the
        // Frobenius norm instead, we scale all error tests by
        // $\sqrt{n}$.
        //
        // A good heuristic is to scale the bound by the square root
        // of the number of floating-point operations.  One could
        // perhaps support this theoretically, since we are using
        // uniform random test problems.
        const magnitude_type fudgeFactor =
          SMT::squareroot(magnitude_type(numRows) *
                          magnitude_type(numCols) *
                          magnitude_type(numCols));
        const magnitude_type TOL = SMT::eps() * fudgeFactor *
          SMT::squareroot(magnitude_type(numCols));

        // Absolute tolerance scaling: the Frobenius norm of the test
        // matrix S.  TOL*ATOL is the absolute tolerance for the
        // residual $\|A - Q*B\|_F$.
        const magnitude_type ATOL = frobeniusNorm (*S);

        sout << "The test matrix S has Frobenius norm " << ATOL
             << ", and the relative error tolerance is TOL = "
             << TOL << "." << endl;

        // Make some copies of S, X1, and X2.  The OrthoManager's
        // project() method shouldn't modify X1 or X2, but this is a a
        // test and we don't know that it doesn't!
        RCP< MV > S_copy = MVT::CloneCopy(*S);
        RCP< MV > Residual = MVT::CloneCopy(*S);
        const int num_X = 2;
        Array< RCP< const MV > > X (num_X);
        X[0] = MVT::CloneCopy(*X1);
        X[1] = MVT::CloneCopy(*X2);

        // Array of coefficients matrices from the projection.
        // For our first test, we allocate each of these matrices
        // with the proper dimensions.
        Array< RCP< mat_type > > C (num_X);
        for (int k = 0; k < num_X; ++k)
          {
            C[k] = rcp (new mat_type (MVT::GetNumberVecs(*X[k]), sizeS));
            C[k]->random(); // will be overwritten
          }
        try {
          // Compute the projection: S_copy := (I - X X^*) S
          OM->project(*S_copy, C, X);

          // Compute the residual: if successful, S = S_copy + X (X^*
          // S =: C) in exact arithmetic.  So, the residual is
          // S - S_copy - X1 C1 - X2 C2.
          //
          // Residual := S - S_copy
          MVT::MvAddMv (SCT::one(), *S, -SCT::one(), *S_copy, *Residual);
          // Residual := Residual - X[k]*C[k]
          for (int k = 0; k < num_X; ++k)
            MVT::MvTimesMatAddMv (-SCT::one(), *X[k], *C[k], SCT::one(), *Residual);
          magnitude_type residErr = frobeniusNorm (*Residual);
          sout << "  ||S - S_copy - X1*C1 - X2*C2||_F = " << residErr;
          if (residErr > ATOL * TOL)
            {
              sout << "  *** Error: ||S - S_copy - X1*C1 - X2*C2||_F = " << residErr
                   << " > ATOL*TOL = " << (ATOL*TOL) << ".";
              numerr++;
            }
          for (int k = 0; k < num_X; ++k)
            {
              // S_copy should be orthogonal to X[k] now.
              const magnitude_type projErr = OM->orthogError(*X[k], *S_copy);
              if (projErr > TOL)
                {
                  sout << "  *** Error: S is not orthogonal to X[" << k
                       << "] by a factor of " << projErr << " > TOL = "
                       << TOL << ".";
                  numerr++;
                }
            }
        } catch (Belos::OrthoError& e) {
          sout << "  *** Error: The OrthoManager subclass instance threw "
            "an exception: " << e.what() << endl;
          numerr++;
        }

        // Print out the collected diagnostic messages, which possibly
        // include error messages.
        const MsgType type = (numerr == 0) ? Debug : static_cast<MsgType> (static_cast<int>(Errors) | static_cast<int>(Debug));
        MyOM->stream(type) << sout.str();
        MyOM->stream(type) << endl;

        return numerr;
      }

      static int
      testProject (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                   const Teuchos::RCP< const MV >& S,
                   const Teuchos::RCP< const MV >& X1,
                   const Teuchos::RCP< const MV >& X2,
                   const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        return testProjectNew (OM, S, X1, X2, MyOM);
      }

      /// Test OrthoManager::project() for the specific OrthoManager instance.
      ///
      /// \return Count of errors (should be zero)
      static int
      testProjectOld (const Teuchos::RCP< Belos::OrthoManager< Scalar, MV > > OM,
                      const Teuchos::RCP< const MV >& S,
                      const Teuchos::RCP< const MV >& X1,
                      const Teuchos::RCP< const MV >& X2,
                      const Teuchos::RCP< Belos::OutputManager< Scalar > >& MyOM)
      {
        using Teuchos::Array;
        using Teuchos::null;
        using Teuchos::RCP;
        using Teuchos::rcp;
        using Teuchos::tuple;

        const scalar_type ONE = SCT::one();
        // We collect all the output in this string wrapper, and print
        // it at the end.
        std::ostringstream sout;
        // Total number of failed tests in this call of this routine.
        int numerr = 0;

        const int numRows = MVT::GetGlobalLength(*S);
        const int numCols = MVT::GetNumberVecs(*S);
        const int sizeS = MVT::GetNumberVecs(*S);
        const int sizeX1 = MVT::GetNumberVecs(*X1);
        const int sizeX2 = MVT::GetNumberVecs(*X2);

        // Relative tolerance against which all tests are performed.
        // We are measuring things in the Frobenius norm $\| \cdot \|_F$.
        // The following bounds hold for all $m \times n$ matrices $A$:
        // \[
        // \|A\|_2 \leq \|A\|_F \leq \sqrt{r} \|A\|_2,
        // \]
        // where $r$ is the (column) rank of $A$.  We bound this above
        // by the number of columns in $A$.
        //
        // Since we are measuring both of these quantitites in the
        // Frobenius norm instead, we scale all error tests by
        // $\sqrt{n}$.
        //
        // A good heuristic is to scale the bound by the square root
        // of the number of floating-point operations.  One could
        // perhaps support this theoretically, since we are using
        // uniform random test problems.
        const magnitude_type fudgeFactor =
          SMT::squareroot(magnitude_type(numRows) *
                          magnitude_type(numCols) *
                          magnitude_type(numCols));
        const magnitude_type TOL = SMT::eps() * fudgeFactor *
          SMT::squareroot(magnitude_type(numCols));

        // Absolute tolerance scaling: the Frobenius norm of the test
        // matrix S.  TOL*ATOL is the absolute tolerance for the
        // residual $\|A - Q*B\|_F$.
        const magnitude_type ATOL = frobeniusNorm (*S);

        sout << "The test matrix S has Frobenius norm " << ATOL
             << ", and the relative error tolerance is TOL = "
             << TOL << "." << endl;


        //
        // Output tests:
        //   <S_out,X1> = 0
        //   <S_out,X2> = 0
        //   S_in = S_out + X1 C1 + X2 C2
        //
        // We will loop over an integer specifying the test combinations.
        // The bit pattern for the different tests is listed in parentheses.
        //
        // For the projectors, test the following combinations:
        // none              (00)
        // P_X1              (01)
        // P_X2              (10)
        // P_X1 P_X2         (11)
        // P_X2 P_X1         (11)
        // The latter two should be tested to give the same result.
        //
        // For each of these, we should test with C1 and C2:
        //
        // if hasM:
        // with and without MX1   (1--)
        // with and without MX2  (1---)
        // with and without MS  (1----)
        //
        // As hasM controls the upper level bits, we need only run test
        // cases 0-3 if hasM==false.  Otherwise, we run test cases 0-31.
        //

        int numtests = 8;

        // test ortho error before orthonormalizing
        if (X1 != null) {
          magnitude_type err = OM->orthogError(*S,*X1);
          sout << "   || <S,X1> || before     : " << err << endl;
        }
        if (X2 != null) {
          magnitude_type err = OM->orthogError(*S,*X2);
          sout << "   || <S,X2> || before     : " << err << endl;
        }

        for (int t = 0; t < numtests; ++t)
          {
            Array< RCP< const MV > > theX;
            Array< RCP< mat_type > > C;
            if ( (t % 3) == 0 ) {
              // neither X1 nor X2
              // C and theX are already empty
            }
            else if ( (t % 3) == 1 ) {
              // X1
              theX = tuple(X1);
              C = tuple( rcp(new mat_type(sizeX1,sizeS)) );
            }
            else if ( (t % 3) == 2 ) {
              // X2
              theX = tuple(X2);
              C = tuple( rcp(new mat_type(sizeX2,sizeS)) );
            }
            else {
              // X1 and X2, and the reverse.
              theX = tuple(X1,X2);
              C = tuple( rcp(new mat_type(sizeX1,sizeS)),
                         rcp(new mat_type(sizeX2,sizeS)) );
            }

            try {
              // call routine
              // if (t && 3) == 3, {
              //    call with reversed input: X2 X1
              // }
              // test all outputs for correctness
              // test all outputs for equivalence

              // here is where the outputs go
              Array< RCP< MV > > S_outs;
              Array< Array< RCP< mat_type > > > C_outs;
              RCP< MV > Scopy;

              // copies of S,MS
              Scopy = MVT::CloneCopy(*S);
              // randomize this data, it should be overwritten
              for (size_type i = 0; i < C.size(); ++i) {
                C[i]->random();
              }
              // Run test.
              // Note that Anasazi and Belos differ, among other places,
              // in the order of arguments to project().
              OM->project(*Scopy,C,theX);
              // we allocate S and MS for each test, so we can save these as views
              // however, save copies of the C
              S_outs.push_back( Scopy );
              C_outs.push_back( Array< RCP< mat_type > >(0) );
              if (C.size() > 0) {
                C_outs.back().push_back( rcp( new mat_type(*C[0]) ) );
              }
              if (C.size() > 1) {
                C_outs.back().push_back( rcp( new mat_type(*C[1]) ) );
              }

              // do we run the reversed input?
              if ( (t % 3) == 3 ) {
                // copies of S,MS
                Scopy = MVT::CloneCopy(*S);
                // randomize this data, it should be overwritten
                for (size_type i = 0; i < C.size(); ++i) {
                  C[i]->random();
                }
                // flip the inputs
                theX = tuple( theX[1], theX[0] );
                // Run test.
                // Note that Anasazi and Belos differ, among other places,
                // in the order of arguments to project().
                OM->project(*Scopy,C,theX);
                // we allocate S and MS for each test, so we can save these as views
                // however, save copies of the C
                S_outs.push_back( Scopy );
                // we are in a special case: P_X1 and P_X2, so we know we applied
                // two projectors, and therefore have two C[i]
                C_outs.push_back( Array<RCP<mat_type > >() );
                // reverse the Cs to compensate for the reverse projectors
                C_outs.back().push_back( rcp( new mat_type(*C[1]) ) );
                C_outs.back().push_back( rcp( new mat_type(*C[0]) ) );
                // flip the inputs back
                theX = tuple( theX[1], theX[0] );
              }

              // test all outputs for correctness
              for (size_type o = 0; o < S_outs.size(); ++o) {
                // S_in = X1*C1 + C2*C2 + S_out
                {
                  RCP<MV> tmp = MVT::CloneCopy(*S_outs[o]);
                  if (C_outs[o].size() > 0) {
                    MVT::MvTimesMatAddMv(ONE,*X1,*C_outs[o][0],ONE,*tmp);
                    if (C_outs[o].size() > 1) {
                      MVT::MvTimesMatAddMv(ONE,*X2,*C_outs[o][1],ONE,*tmp);
                    }
                  }
                  magnitude_type err = MVDiff(*tmp,*S);
                  if (err > ATOL*TOL) {
                    sout << "         vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv         tolerance exceeded! test failed!" << endl;
                    numerr++;
                  }
                  sout << "  " << t << "|| S_in - X1*C1 - X2*C2 - S_out || : " << err << endl;
                }
                // <X1,S> == 0
                if (theX.size() > 0 && theX[0] != null) {
                  magnitude_type err = OM->orthogError(*theX[0],*S_outs[o]);
                  if (err > TOL) {
                    sout << "         vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv         tolerance exceeded! test failed!" << endl;
                    numerr++;
                  }
                  sout << "  " << t << "|| <X[0],S> || after      : " << err << endl;
                }
                // <X2,S> == 0
                if (theX.size() > 1 && theX[1] != null) {
                  magnitude_type err = OM->orthogError(*theX[1],*S_outs[o]);
                  if (err > TOL) {
                    sout << "         vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv         tolerance exceeded! test failed!" << endl;
                    numerr++;
                  }
                  sout << "  " << t << "|| <X[1],S> || after      : " << err << endl;
                }
              }

              // test all outputs for equivalence
              // check all combinations:
              //    output 0 == output 1
              //    output 0 == output 2
              //    output 1 == output 2
              for (size_type o1=0; o1<S_outs.size(); o1++) {
                for (size_type o2=o1+1; o2<S_outs.size(); o2++) {
                  // don't need to check MS_outs because we check
                  //   S_outs and MS_outs = M*S_outs
                  // don't need to check C_outs either
                  //
                  // check that S_outs[o1] == S_outs[o2]
                  magnitude_type err = MVDiff(*S_outs[o1],*S_outs[o2]);
                  if (err > TOL) {
                    sout << "    vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv         tolerance exceeded! test failed!" << endl;
                    numerr++;
                  }
                }
              }

            }
            catch (Belos::OrthoError& e) {
              sout << "   -------------------------------------------         project() threw exception" << endl;
              sout << "   Error: " << e.what() << endl;
              numerr++;
            }
          } // test for

        MsgType type = Debug;
        if (numerr>0) type = Errors;
        MyOM->stream(type) << sout.str();
        MyOM->stream(type) << endl;

        return numerr;
      }


    };



  } // namespace Test
} // namespace Belos