- Introduction        las2: sparse svd via single-vector Lanczos algorithm with              selective re-orthogonalization.	las2.c is an ANSI-C code designed to find some eigenvalues	and eigenvectors of a linear operator opb that is real	symmetric. This codes is a re-write of the lanso code (in        Fortran-77) distributed by B. Parlett and his colleagues at        UC-Berkeley.          In las2.c, the operator B is assumed to be of the form                 B =  A'A,   where A is m by n (m>>n) and sparse.        Hence, the singular triplets of A are computed as the eigenpairs        of B.  The eigenvalues of B are the squares of the singular values of        A, the eigenvectors correspond to the right singular vectors only.        The left singular vectors of A are then determined by                          u = 1/sigma A*v,        where {u,sigma,v} is a singular triplet of A.	las2.c implements the simple lanczos algorithm with Simon's        selective orthogonalization to actively maintain extended        semi-orthogonality amongst the computed Lanczos vectors.	These methods are most efficient when only a modest fraction        (less than 1/4th) of the B-eigenpairs are wanted.  One potentially         useful feature of both codes are their delivery of certain	certain eigenvalues not yet good to full precision, for some 	tasks they are often good enough.	The largest eigenvalues of B tend to be captured before the small ones	but convergence depends on separation ratios and these vary with	the application.   las2.c will find eigenvalues of multiplicity        greater than one, but the copies stabilize one at a time, the         interval between copies depends on the problem and wordlength.         Experience suggests the gap is between 4 and 20 steps.- User-supplied routines        For las2.c, the user must specify multiplication by both        matrix B and A only (subroutines opb and opa, respectively).	The specification of opb should look something like                void opb(long n,double *x, double *y);	so that opb takes a vector x and returns y = B*x, where        B is the appropriate matrix (see above).	The specification of opa should look something like                void opa(double *x, double *y);	so that opa takes a vector x and returns y = A*x.	Subroutine opb will be called with n always equal to the	dimension of the eigenproblem solved. In las2.c we use        the Harwell-Boeing sparse matrix format for accessing        elements of the nrow by ncol sparse matrix A and its        transpose (denoted A').  Other sparse matrix formats can be         used, of course.	If secondary storage (such as a disk drive) must be used to	store the computed Lanczos vectors, subroutine "store"	must be supplied by the user in place of our trivially-	implemented store which assumes the existence of virtual	memory.  The specification of store in las2.c is	                void store(long n, long isw, long j, double *s);	 	where	n	... the dimension of the eigenproblem (ncol, for nrow>>ncol),	isw	... action type switch, can only be 1, 2, 3 or 4.		    isw = 1 requests storing of j-th lanczos vector q(j),		    isw = 2 requests retrieval of j-th lanczos vector q(j),		    isw = 3 requests storing of m*q(j) for j = 1 or 2,		    isw = 4 requests retrieval of m*q(j) for j = 1 or 2.	j	... index of the j-th lanczos vector,	s	... the n-vector to be stored/retrieved.- Termination 	las2.c will terminate if either	1) maxprs eigenpairs have been found, or	2) lanmax lanczos steps have been taken, or	3) two ritz values inside [endl,endr] have stabilized.	whichever occurs first.  Note that an eigenvalue of multiplicity	higher than one may not have all its copies delivered.- Acknowledgements	Beresford Parlett and his colleagues (David Scott, Horst Simon,	Bahram Nour-Omid, John Lewis, and Jeremy Du Croz) are to be credited	for the LANSO strategy and initial Fortran-77 code from which        las1.c and las2.c eventually evolved.- Information         Please address all questions, comments, or corrections to:        M. W. Berry        Department of Computer Science        University of Tennessee        107 Ayres Hall        Knoxville, TN  37996-1301        email: berry@cs.utk.edu        phone: (615) 974-5067-File descriptions        las2.c requires the include file las1.h for compilation.        The input and output files associated with las2.c are        listed below.             Code           Input         Output            ------      ------------    ---------            las2.c      lap2, matrix    lao2,lav2       The binary output files lav2 (which contains the        approximate right singular vectors followed by the ap-       proximate left singular vectors) will be created by       las2.c if it does not already exist.  If you are       running on a Unix-based workstation you should uncomment       the line                 /*   #define  UNIX_CREAT */       in the declarations prior to main() in las2.c.       UNIX_CREAT specifies the use of the UNIX "creat" system       routine with the permissions defined by the PERMS constant                  #define PERMS 0664       You may adjust PERMS for the desired permissions on the       lav2 file (default is Read/Write for user and group,       and Read for others).  Subsequent runs will be able to       open and overwrite these files with the default permissions.       las2.c obtains its parameters specifying the       sparse SVD problem to be solved from the input file       lap2.  This parameter files contains the single line	 <name> lanmax maxprs endl endr vectors kappa       where         <name>     is the name of the data set;         lanmax     is an integer specifying maximum number of lanczos                   step allowed;        maxprs     indicates  maximum number of singular triplets of A                    (eigenpairs of the equivalent matrix B) desired;         endl,endr  are two end-points of an interval within which all                    unwanted eigenvalues of the particular matrix B lie;         vectors    contains the string TRUE or FALSE to indicate when                    singular triplets are needed (TRUE) and when only                    singular values are needed (FALSE);        kappa      contains the relative accuracy of ritz values                    acceptable as eigenvalues of the matrix B.        - Sparse matrix format        las2.c is designed to read input matrices that are stored        in the Harwell-Boeing sparse matrix format.  The nonzeros        of such matrices are stored in a compressed column-oriented        format.  The row indices and corresponding nonzero values        are stored by columns with a column start index array        whose entries contain pointers to the nonzero starting each        column.  las2.c reads the sparse matrix data from the input        file called "matrix".        Each input file "matrix" should begin with a four-line header        record followed by three more records containing, in order,         the column-start pointers, the row indices, and the nonzero        numerical values.        The first line of the header consists of a 72-character title        and an 8-character key by which the matrices are referenced.        The second line can be used for comments or to indicate record        length for each index or value array.  Although this line is         generally ignored, A CHARACTER MUST BE PLACED ON THAT LINE.        The third line contains a three-character string denoting the        matrix type and the three integers specifying the number of rows,        columns, and nonzeros.  The fourth line which usually contains        input format for Fortran-77 I/O is ignored by our ANSI-C code.        The exact format is		"%72c %*s %*s %*s %d %d %d %*d"	for the first three lines of the header,		line 1      <title>         <key>		 	(col.  1 - 72) (col. 73 - 80)		line 2   <string>		line 3   <matrix type> nrow ncol nnzero 	and 		"%*s %*s %*s %*s"	for the last line of the header.		line 4   <string1> <string2> <string3> <string4>        Even though only the title and the integers specifying the        number of rows, columns, and nonzero elements are read, other        strings of input must be present in indicated positions.        Otherwise, the format of the "fscanf" statements must be         changed accordingly.-Recommendation        Keep in mind that matrix B is of the form (' denotes transposition)           B= [O  A], for las1.c and  B= A'A for las2.c.              [A' O]        When ill-conditioning is not likely, we recommend the use        of las2.c (smaller eigensystems and less memory).  Otherwise,        we suggest that you use las1.c which is a root-free method        yet approximates +/- pairs of each singular value of A.