DCHKST2STG  checks the symmetric eigenvalue problem routines


 using the 2-stage reduction techniques. Since the generation


 of Q or the vectors is not available in this release, we only 


 compare the eigenvalue resulting when using the 2-stage to the 


 one considered as reference using the standard 1-stage reduction


 DSYTRD. For that, we call the standard DSYTRD and compute D1 using 


 DSTEQR, then we call the 2-stage DSYTRD_2STAGE with Upper and Lower


 and we compute D2 and D3 using DSTEQR and then we replaced tests


 3 and 4 by tests 11 and 12. test 1 and 2 remain to verify that 


 the 1-stage results are OK and can be trusted.


 This testing routine will converge to the DCHKST in the next 


 release when vectors and generation of Q will be implemented.


    DSYTRD factors A as  U S U' , where ' means transpose,


    S is symmetric tridiagonal, and U is orthogonal.


    DSYTRD can use either just the lower or just the upper triangle


    of A; DCHKST2STG checks both cases.


    U is represented as a product of Householder


    transformations, whose vectors are stored in the first


    n-1 columns of V, and whose scale factors are in TAU.


    DSPTRD does the same as DSYTRD, except that A and V are stored


    in 'packed' format.


    DORGTR constructs the matrix U from the contents of V and TAU.


    DOPGTR constructs the matrix U from the contents of VP and TAU.


    DSTEQR factors S as  Z D1 Z' , where Z is the orthogonal


    matrix of eigenvectors and D1 is a diagonal matrix with


    the eigenvalues on the diagonal.  D2 is the matrix of


    eigenvalues computed when Z is not computed.


    DSTERF computes D3, the matrix of eigenvalues, by the


    PWK method, which does not yield eigenvectors.


    DPTEQR factors S as  Z4 D4 Z4' , for a


    symmetric positive definite tridiagonal matrix.


    D5 is the matrix of eigenvalues computed when Z is not


    computed.


    DSTEBZ computes selected eigenvalues.  WA1, WA2, and


    WA3 will denote eigenvalues computed to high


    absolute accuracy, with different range options.


    WR will denote eigenvalues computed to high relative


    accuracy.


    DSTEIN computes Y, the eigenvectors of S, given the


    eigenvalues.


    DSTEDC factors S as Z D1 Z' , where Z is the orthogonal


    matrix of eigenvectors and D1 is a diagonal matrix with


    the eigenvalues on the diagonal ('I' option). It may also


    update an input orthogonal matrix, usually the output


    from DSYTRD/DORGTR or DSPTRD/DOPGTR ('V' option). It may


    also just compute eigenvalues ('N' option).


    DSTEMR factors S as Z D1 Z' , where Z is the orthogonal


    matrix of eigenvectors and D1 is a diagonal matrix with


    the eigenvalues on the diagonal ('I' option).  DSTEMR


    uses the Relatively Robust Representation whenever possible.


 When DCHKST2STG is called, a number of matrix 'sizes' ('n's') and a


 number of matrix 'types' are specified.  For each size ('n')


 and each type of matrix, one matrix will be generated and used


 to test the symmetric eigenroutines.  For each matrix, a number


 of tests will be performed:


 (1)     | A - V S V' | / ( |A| n ulp ) DSYTRD( UPLO='U', ... )


 (2)     | I - UV' | / ( n ulp )        DORGTR( UPLO='U', ... )


 (3)     | A - V S V' | / ( |A| n ulp ) DSYTRD( UPLO='L', ... )


         replaced by | D1 - D2 | / ( |D1| ulp ) where D1 is the 


         eigenvalue matrix computed using S and D2 is the 


         eigenvalue matrix computed using S_2stage the output of


         DSYTRD_2STAGE('N', 'U',....). D1 and D2 are computed 


         via DSTEQR('N',...)  


 (4)     | I - UV' | / ( n ulp )        DORGTR( UPLO='L', ... )


         replaced by | D1 - D3 | / ( |D1| ulp ) where D1 is the 


         eigenvalue matrix computed using S and D3 is the 


         eigenvalue matrix computed using S_2stage the output of


         DSYTRD_2STAGE('N', 'L',....). D1 and D3 are computed 


         via DSTEQR('N',...)  


 (5-8)   Same as 1-4, but for DSPTRD and DOPGTR.


 (9)     | S - Z D Z' | / ( |S| n ulp ) DSTEQR('V',...)


 (10)    | I - ZZ' | / ( n ulp )        DSTEQR('V',...)


 (11)    | D1 - D2 | / ( |D1| ulp )        DSTEQR('N',...)


 (12)    | D1 - D3 | / ( |D1| ulp )        DSTERF


 (13)    0 if the true eigenvalues (computed by sturm count)


         of S are within THRESH of


         those in D1.  2*THRESH if they are not.  (Tested using


         DSTECH)


 For S positive definite,


 (14)    | S - Z4 D4 Z4' | / ( |S| n ulp ) DPTEQR('V',...)


 (15)    | I - Z4 Z4' | / ( n ulp )        DPTEQR('V',...)


 (16)    | D4 - D5 | / ( 100 |D4| ulp )       DPTEQR('N',...)


 When S is also diagonally dominant by the factor gamma < 1,


 (17)    max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,


          i


         omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4


                                              DSTEBZ( 'A', 'E', ...)


 (18)    | WA1 - D3 | / ( |D3| ulp )          DSTEBZ( 'A', 'E', ...)


 (19)    ( max { min | WA2(i)-WA3(j) | } +


            i     j


           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )


            i     j


                                              DSTEBZ( 'I', 'E', ...)


 (20)    | S - Y WA1 Y' | / ( |S| n ulp )  DSTEBZ, SSTEIN


 (21)    | I - Y Y' | / ( n ulp )          DSTEBZ, SSTEIN


 (22)    | S - Z D Z' | / ( |S| n ulp )    DSTEDC('I')


 (23)    | I - ZZ' | / ( n ulp )           DSTEDC('I')


 (24)    | S - Z D Z' | / ( |S| n ulp )    DSTEDC('V')


 (25)    | I - ZZ' | / ( n ulp )           DSTEDC('V')


 (26)    | D1 - D2 | / ( |D1| ulp )           DSTEDC('V') and


                                              DSTEDC('N')


 Test 27 is disabled at the moment because DSTEMR does not


 guarantee high relatvie accuracy.


 (27)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,


          i


         omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4


                                              DSTEMR('V', 'A')


 (28)    max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,


          i


         omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4


                                              DSTEMR('V', 'I')


 Tests 29 through 34 are disable at present because DSTEMR


 does not handle partial spectrum requests.


 (29)    | S - Z D Z' | / ( |S| n ulp )    DSTEMR('V', 'I')


 (30)    | I - ZZ' | / ( n ulp )           DSTEMR('V', 'I')


 (31)    ( max { min | WA2(i)-WA3(j) | } +


            i     j


           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )


            i     j


         DSTEMR('N', 'I') vs. SSTEMR('V', 'I')


 (32)    | S - Z D Z' | / ( |S| n ulp )    DSTEMR('V', 'V')


 (33)    | I - ZZ' | / ( n ulp )           DSTEMR('V', 'V')


 (34)    ( max { min | WA2(i)-WA3(j) | } +


            i     j


           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )


            i     j


         DSTEMR('N', 'V') vs. SSTEMR('V', 'V')


 (35)    | S - Z D Z' | / ( |S| n ulp )    DSTEMR('V', 'A')


 (36)    | I - ZZ' | / ( n ulp )           DSTEMR('V', 'A')


 (37)    ( max { min | WA2(i)-WA3(j) | } +


            i     j


           max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )


            i     j


         DSTEMR('N', 'A') vs. SSTEMR('V', 'A')


 The 'sizes' are specified by an array NN(1:NSIZES); the value of


 each element NN(j) specifies one size.


 The 'types' are specified by a logical array DOTYPE( 1:NTYPES );


 if DOTYPE(j) is .TRUE., then matrix type 'j' will be generated.


 Currently, the list of possible types is:


 (1)  The zero matrix.


 (2)  The identity matrix.


 (3)  A diagonal matrix with evenly spaced entries


      1, ..., ULP  and random signs.


      (ULP = (first number larger than 1) - 1 )


 (4)  A diagonal matrix with geometrically spaced entries


      1, ..., ULP  and random signs.


 (5)  A diagonal matrix with 'clustered' entries 1, ULP, ..., ULP


      and random signs.


 (6)  Same as (4), but multiplied by SQRT( overflow threshold )


 (7)  Same as (4), but multiplied by SQRT( underflow threshold )


 (8)  A matrix of the form  U' D U, where U is orthogonal and


      D has evenly spaced entries 1, ..., ULP with random signs


      on the diagonal.


 (9)  A matrix of the form  U' D U, where U is orthogonal and


      D has geometrically spaced entries 1, ..., ULP with random


      signs on the diagonal.


 (10) A matrix of the form  U' D U, where U is orthogonal and


      D has 'clustered' entries 1, ULP,..., ULP with random


      signs on the diagonal.


 (11) Same as (8), but multiplied by SQRT( overflow threshold )


 (12) Same as (8), but multiplied by SQRT( underflow threshold )


 (13) Symmetric matrix with random entries chosen from (-1,1).


 (14) Same as (13), but multiplied by SQRT( overflow threshold )


 (15) Same as (13), but multiplied by SQRT( underflow threshold )


 (16) Same as (8), but diagonal elements are all positive.


 (17) Same as (9), but diagonal elements are all positive.


 (18) Same as (10), but diagonal elements are all positive.


 (19) Same as (16), but multiplied by SQRT( overflow threshold )


 (20) Same as (16), but multiplied by SQRT( underflow threshold )


 (21) A diagonally dominant tridiagonal matrix with geometrically


      spaced diagonal entries 1, ..., ULP.

NSIZES

          NSIZES is INTEGER


          The number of sizes of matrices to use.  If it is zero,


          DCHKST2STG does nothing.  It must be at least zero.

NN

          NN is INTEGER array, dimension (NSIZES)


          An array containing the sizes to be used for the matrices.


          Zero values will be skipped.  The values must be at least


          zero.

NTYPES

          NTYPES is INTEGER


          The number of elements in DOTYPE.   If it is zero, DCHKST2STG


          does nothing.  It must be at least zero.  If it is MAXTYP+1


          and NSIZES is 1, then an additional type, MAXTYP+1 is


          defined, which is to use whatever matrix is in A.  This


          is only useful if DOTYPE(1:MAXTYP) is .FALSE. and


          DOTYPE(MAXTYP+1) is .TRUE. .

DOTYPE

          DOTYPE is LOGICAL array, dimension (NTYPES)


          If DOTYPE(j) is .TRUE., then for each size in NN a


          matrix of that size and of type j will be generated.


          If NTYPES is smaller than the maximum number of types


          defined (PARAMETER MAXTYP), then types NTYPES+1 through


          MAXTYP will not be generated.  If NTYPES is larger


          than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)


          will be ignored.

ISEED

          ISEED is INTEGER array, dimension (4)


          On entry ISEED specifies the seed of the random number


          generator. The array elements should be between 0 and 4095;


          if not they will be reduced mod 4096.  Also, ISEED(4) must


          be odd.  The random number generator uses a linear


          congruential sequence limited to small integers, and so


          should produce machine independent random numbers. The


          values of ISEED are changed on exit, and can be used in the


          next call to DCHKST2STG to continue the same random number


          sequence.

THRESH

          THRESH is DOUBLE PRECISION


          A test will count as 'failed' if the 'error', computed as


          described above, exceeds THRESH.  Note that the error


          is scaled to be O(1), so THRESH should be a reasonably


          small multiple of 1, e.g., 10 or 100.  In particular,


          it should not depend on the precision (single vs. double)


          or the size of the matrix.  It must be at least zero.

NOUNIT

          NOUNIT is INTEGER


          The FORTRAN unit number for printing out error messages


          (e.g., if a routine returns IINFO not equal to 0.)

A

          A is DOUBLE PRECISION array of


                                  dimension ( LDA , max(NN) )


          Used to hold the matrix whose eigenvalues are to be


          computed.  On exit, A contains the last matrix actually


          used.

LDA

          LDA is INTEGER


          The leading dimension of A.  It must be at


          least 1 and at least max( NN ).

AP

          AP is DOUBLE PRECISION array of


                      dimension( max(NN)*max(NN+1)/2 )


          The matrix A stored in packed format.

SD

          SD is DOUBLE PRECISION array of


                             dimension( max(NN) )


          The diagonal of the tridiagonal matrix computed by DSYTRD.


          On exit, SD and SE contain the tridiagonal form of the


          matrix in A.

SE

          SE is DOUBLE PRECISION array of


                             dimension( max(NN) )


          The off-diagonal of the tridiagonal matrix computed by


          DSYTRD.  On exit, SD and SE contain the tridiagonal form of


          the matrix in A.

D1

          D1 is DOUBLE PRECISION array of


                             dimension( max(NN) )


          The eigenvalues of A, as computed by DSTEQR simultaneously


          with Z.  On exit, the eigenvalues in D1 correspond with the


          matrix in A.

D2

          D2 is DOUBLE PRECISION array of


                             dimension( max(NN) )


          The eigenvalues of A, as computed by DSTEQR if Z is not


          computed.  On exit, the eigenvalues in D2 correspond with


          the matrix in A.

D3

          D3 is DOUBLE PRECISION array of


                             dimension( max(NN) )


          The eigenvalues of A, as computed by DSTERF.  On exit, the


          eigenvalues in D3 correspond with the matrix in A.

D4

          D4 is DOUBLE PRECISION array of


                             dimension( max(NN) )


          The eigenvalues of A, as computed by DPTEQR(V).


          DPTEQR factors S as  Z4 D4 Z4*


          On exit, the eigenvalues in D4 correspond with the matrix in A.

D5

          D5 is DOUBLE PRECISION array of


                             dimension( max(NN) )


          The eigenvalues of A, as computed by DPTEQR(N)


          when Z is not computed. On exit, the


          eigenvalues in D4 correspond with the matrix in A.

WA1

          WA1 is DOUBLE PRECISION array of


                             dimension( max(NN) )


          All eigenvalues of A, computed to high


          absolute accuracy, with different range options.


          as computed by DSTEBZ.

WA2

          WA2 is DOUBLE PRECISION array of


                             dimension( max(NN) )


          Selected eigenvalues of A, computed to high


          absolute accuracy, with different range options.


          as computed by DSTEBZ.


          Choose random values for IL and IU, and ask for the


          IL-th through IU-th eigenvalues.

WA3

          WA3 is DOUBLE PRECISION array of


                             dimension( max(NN) )


          Selected eigenvalues of A, computed to high


          absolute accuracy, with different range options.


          as computed by DSTEBZ.


          Determine the values VL and VU of the IL-th and IU-th


          eigenvalues and ask for all eigenvalues in this range.

WR

          WR is DOUBLE PRECISION array of


                             dimension( max(NN) )


          All eigenvalues of A, computed to high


          absolute accuracy, with different options.


          as computed by DSTEBZ.

U

          U is DOUBLE PRECISION array of


                             dimension( LDU, max(NN) ).


          The orthogonal matrix computed by DSYTRD + DORGTR.

LDU

          LDU is INTEGER


          The leading dimension of U, Z, and V.  It must be at least 1


          and at least max( NN ).

V

          V is DOUBLE PRECISION array of


                             dimension( LDU, max(NN) ).


          The Housholder vectors computed by DSYTRD in reducing A to


          tridiagonal form.  The vectors computed with UPLO='U' are


          in the upper triangle, and the vectors computed with UPLO='L'


          are in the lower triangle.  (As described in DSYTRD, the


          sub- and superdiagonal are not set to 1, although the


          true Householder vector has a 1 in that position.  The


          routines that use V, such as DORGTR, set those entries to


          1 before using them, and then restore them later.)

VP

          VP is DOUBLE PRECISION array of


                      dimension( max(NN)*max(NN+1)/2 )


          The matrix V stored in packed format.

TAU

          TAU is DOUBLE PRECISION array of


                             dimension( max(NN) )


          The Householder factors computed by DSYTRD in reducing A


          to tridiagonal form.

Z

          Z is DOUBLE PRECISION array of


                             dimension( LDU, max(NN) ).


          The orthogonal matrix of eigenvectors computed by DSTEQR,


          DPTEQR, and DSTEIN.

WORK

          WORK is DOUBLE PRECISION array of


                      dimension( LWORK )

LWORK

          LWORK is INTEGER


          The number of entries in WORK.  This must be at least


          1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2


          where Nmax = max( NN(j), 2 ) and lg = log base 2.

IWORK

          IWORK is INTEGER array,


          Workspace.

LIWORK

          LIWORK is INTEGER


          The number of entries in IWORK.  This must be at least


                  6 + 6*Nmax + 5 * Nmax * lg Nmax


          where Nmax = max( NN(j), 2 ) and lg = log base 2.

RESULT

          RESULT is DOUBLE PRECISION array, dimension (26)


          The values computed by the tests described above.


          The values are currently limited to 1/ulp, to avoid


          overflow.

INFO

          INFO is INTEGER


          If 0, then everything ran OK.


           -1: NSIZES < 0


           -2: Some NN(j) < 0


           -3: NTYPES < 0


           -5: THRESH < 0


           -9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).


          -23: LDU < 1 or LDU < NMAX.


          -29: LWORK too small.


          If  DLATMR, SLATMS, DSYTRD, DORGTR, DSTEQR, SSTERF,


              or DORMC2 returns an error code, the


              absolute value of it is returned.
-----------------------------------------------------------------------


       Some Local Variables and Parameters:


       ---- ----- --------- --- ----------


       ZERO, ONE       Real 0 and 1.


       MAXTYP          The number of types defined.


       NTEST           The number of tests performed, or which can


                       be performed so far, for the current matrix.


       NTESTT          The total number of tests performed so far.


       NBLOCK          Blocksize as returned by ENVIR.


       NMAX            Largest value in NN.


       NMATS           The number of matrices generated so far.


       NERRS           The number of tests which have exceeded THRESH


                       so far.


       COND, IMODE     Values to be passed to the matrix generators.


       ANORM           Norm of A; passed to matrix generators.


       OVFL, UNFL      Overflow and underflow thresholds.


       ULP, ULPINV     Finest relative precision and its inverse.


       RTOVFL, RTUNFL  Square roots of the previous 2 values.


               The following four arrays decode JTYPE:


       KTYPE(j)        The general type (1-10) for type 'j'.


       KMODE(j)        The MODE value to be passed to the matrix


                       generator for type 'j'.


       KMAGN(j)        The order of magnitude ( O(1),


                       O(overflow^(1/2) ), O(underflow^(1/2) )

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.