ZCHKST2STG(3)

📦
version 3.12.1 · 2 versions
, in arch rolling

okay bestie, welcome to ZCHKST2STG(3) — the page for TESTING/EIG/zchkst2stg.f, living in the Library Functions Manual. It’s giving testing energy, and no cap, we’re keeping every single character of these signatures. 💅

🌸 NAME 🌸

TESTING/EIG/zchkst2stg.f — that’s the file name, bestie, and we’re not changing a single character of it. Periodt.

✨ SYNOPSIS ✨

🧚 Functions/Subroutines 🧚

okay so under Functions/Subroutines, the synopsis is literally this call:

subroutine zchkst2stg (nsizes, nn, ntypes, dotype, iseed, thresh,
    nounit, a, lda, ap, sd, se, d1, d2, d3, d4, d5, wa1, wa2, wa3, wr,
    u, ldu, v, vp, tau, z, work, lwork, rwork, lrwork, iwork, liwork,
    result, info)

ZCHKST2STG

💅 Function/Subroutine Documentation 💅

now the full documentation signature — it’s a long one, and every arg is here, exact, no edits:

subroutine zchkst2stg (integer nsizes, integer, dimension( * ) nn, integer
       ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
       double precision thresh, integer nounit, complex*16, dimension( lda, * )
       a, integer lda, complex*16, dimension( * ) ap, double precision,
       dimension( * ) sd, double precision, dimension( * ) se, double
       precision, dimension( * ) d1, double precision, dimension( * ) d2,
       double precision, dimension( * ) d3, double precision, dimension( * )
       d4, double precision, dimension( * ) d5, double precision, dimension( *
       ) wa1, double precision, dimension( * ) wa2, double precision,
       dimension( * ) wa3, double precision, dimension( * ) wr, complex*16,
       dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldu, * ) v,
       complex*16, dimension( * ) vp, complex*16, dimension( * ) tau,
       complex*16, dimension( ldu, * ) z, complex*16, dimension( * ) work,
       integer lwork, double precision, dimension( * ) rwork, integer lrwork,
       integer, dimension( * ) iwork, integer liwork, double precision,
       dimension( * ) result, integer info)
       ZCHKST2STG

ZCHKST2STG

🎯 Purpose

okay so here's the tea on what this bad boy actually checks, because the whole point of ZCHKST2STG is verifying Hermitian eigenvalue problem routines using the 2-stage reduction techniques 🔥

So here's the situation, bestie: 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 ZHETRD. For that, we call the standard ZHETRD and compute D1 using DSTEQR, then we call the 2-stage ZHETRD_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 ZCHKST in the next release when vectors and generation of Q will be implemented. 👑

🧚 the cast of characters (aka what's being tested)

  • ZHETRD factors A as U S U* , where * means conjugate transpose, S is real symmetric tridiagonal, and U is unitary. ZHETRD can use either just the lower or just the upper triangle of A; ZCHKST2STG 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.
  • ZHPTRD does the same as ZHETRD, except that A and V are stored in format. (yep, that blank is literally in the source, bestie — the format name got eaten, not me)
  • ZUNGTR constructs the matrix U from the contents of V and TAU.
  • ZUPGTR constructs the matrix U from the contents of VP and TAU.
  • ZSTEQR factors S as Z D1 Z* , where Z is the unitary 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.
  • ZPTEQR factors S as Z4 D4 Z4* , for a Hermitian 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.
  • ZSTEIN computes Y, the eigenvectors of S, given the eigenvalues.
  • ZSTEDC factors S as Z D1 Z* , where Z is the unitary matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal ('I' option). It may also update an input unitary matrix, usually the output from ZHETRD/ZUNGTR or ZHPTRD/ZUPGTR ('V' option). It may also just compute eigenvalues ('N' option).
  • ZSTEMR factors S as Z D1 Z* , where Z is the unitary matrix of eigenvectors and D1 is a diagonal matrix with the eigenvalues on the diagonal ('I' option). ZSTEMR uses the Relatively Robust Representation whenever possible. ✨

🎀 the setup, explained

When ZCHKST2STG is called, a number of matrix () and a number of matrix are specified. For each size () and each type of matrix, one matrix will be generated and used to test the Hermitian eigenroutines. For each matrix, a number of tests will be performed: (this is the fun part, fr fr)

🦋 THE TESTS — all of them, no skips

  • (1) | A - V S V* | / ( |A| n ulp ) — ZHETRD( UPLO='U', ... )
  • (2) | I - UV* | / ( n ulp ) — ZUNGTR( UPLO='U', ... )
  • (3) | A - V S V* | / ( |A| n ulp ) — ZHETRD( 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 ZHETRD_2STAGE(, ,....). D1 and D2 are computed via DSTEQR('N',...)
  • (4) | I - UV* | / ( n ulp ) — ZUNGTR( 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 ZHETRD_2STAGE(, ,....). D1 and D3 are computed via DSTEQR('N',...)
  • (5-8) Same as 1-4, but for ZHPTRD and ZUPGTR.
  • (9) | S - Z D Z* | / ( |S| n ulp ) — ZSTEQR('V',...)
  • (10) | I - ZZ* | / ( n ulp ) — ZSTEQR('V',...)
  • (11) | D1 - D2 | / ( |D1| ulp ) — ZSTEQR('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 ) — ZPTEQR('V',...)
  • (15) | I - Z4 Z4* | / ( n ulp ) — ZPTEQR('V',...)
  • (16) | D4 - D5 | / ( 100 |D4| ulp ) — ZPTEQR('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, ZSTEIN
  • (21) | I - Y Y* | / ( n ulp ) — DSTEBZ, ZSTEIN
  • (22) | S - Z D Z* | / ( |S| n ulp ) — ZSTEDC('I')
  • (23) | I - ZZ* | / ( n ulp ) — ZSTEDC('I')
  • (24) | S - Z D Z* | / ( |S| n ulp ) — ZSTEDC('V')
  • (25) | I - ZZ* | / ( n ulp ) — ZSTEDC('V')
  • (26) | D1 - D2 | / ( |D1| ulp ) — ZSTEDC('V') and ZSTEDC('N')
heads up, bestie: Test 27 is disabled at the moment because ZSTEMR does not guarantee high relative accuracy. 😭
  • (27)
max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
 i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                     ZSTEMR('V', 'A')
  • (28)
max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
 i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                     ZSTEMR('V', 'I')
also very much disabled rn: Tests 29 through 34 are disable at present because ZSTEMR does not handle partial spectrum requests. 💀
  • (29) | S - Z D Z* | / ( |S| n ulp ) — ZSTEMR('V', 'I')
  • (30) | I - ZZ* | / ( n ulp ) — ZSTEMR('V', 'I')
  • (31)
( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j
ZSTEMR('N', 'I') vs. CSTEMR('V', 'I')
  • (32) | S - Z D Z* | / ( |S| n ulp ) — ZSTEMR('V', 'V')
  • (33) | I - ZZ* | / ( n ulp ) — ZSTEMR('V', 'V')
  • (34)
( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j
ZSTEMR('N', 'V') vs. CSTEMR('V', 'V')
  • (35) | S - Z D Z* | / ( |S| n ulp ) — ZSTEMR('V', 'A')
  • (36) | I - ZZ* | / ( n ulp ) — ZSTEMR('V', 'A')
  • (37)
( max { min | WA2(i)-WA3(j) | } +
   i     j
  max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
   i     j
ZSTEMR('N', 'A') vs. CSTEMR('V', 'A')

🌸 the sizes and the matrix types

The are specified by an array NN(1:NSIZES); the value of each element NN(j) specifies one size.

The are specified by a logical array DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type will be generated.

Currently, the list of possible types is: (each one is a whole vibe, ngl)

  • (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 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 unitary 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 unitary 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 unitary and D has 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) Hermitian 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.

and that's the whole matrix-type menu, bestie — 21 flavours of test matrix, all iconic, all accounted for 💅

✨ Parameters ✨

Okay bestie, here comes the full cast of characters 💅 every single one of these is a real argument you gotta pass, and I'm not skipping a single one. Buckle up.

  • NSIZES — INTEGER. It's the number of sizes of matrices to use. If it's zero, ZCHKST2STG does nothing (iconic king of boundaries, honestly). Must be at least zero.
  • NN — INTEGER array, dimension (NSIZES). The array holding the sizes to be used for the matrices. Zero values get skipped, no cap. The values must be at least zero.
  • NTYPES — INTEGER. The number of elements in DOTYPE. If it's zero, ZCHKST2STG does nothing. It must be at least zero. If it equals MAXTYP+1 and NSIZES is 1, then one extra type gets defined, MAXTYP+1, and that one means "just use whatever matrix is already in A." This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE..
  • DOTYPE — LOGICAL array, dimension (NTYPES). If DOTYPE(j) is .TRUE., then for every size in NN a matrix of that size and of type j gets generated. If NTYPES is smaller than the maximum number of types defined (PARAMETER MAXTYP), then types NTYPES+1 through MAXTYP simply do not get generated. If NTYPES is larger than MAXTYP, then DOTYPE(MAXTYP+1) through DOTYPE(NTYPES) get ignored. 🫶
  • ISEED — INTEGER array, dimension (4). On entry, ISEED sets the seed of the random number generator. The array elements should be between 0 and 4095; if they're not, they get reduced mod 4096. Also, ISEED(4) must be odd — not optional, bestie, it's a whole vibe requirement. The RNG uses a linear congruential sequence limited to small integers, so it should crank out machine independent random numbers. The values of ISEED are changed on exit, which means you can feed them into the next call to ZCHKST2STG to keep the exact same random number sequence going. Continuity slay ✨
  • THRESH — DOUBLE PRECISION. A test will count as ⬜ if the ⬜ — computed as described above — exceeds THRESH. (Yeah, the source doc literally has blanks there. Not me. Blame the Fortran 💀.) Note that the error is scaled to be O(1), so THRESH should be a reasonably small multiple of 1, like, say, 10 or 100. In particular it should not depend on the precision (single vs. double) or on the size of the matrix. It must be at least zero.
  • NOUNIT — INTEGER. The FORTRAN unit number for printing out error messages (like, if a routine returns IINFO not equal to 0).
  • A — COMPLEX*16 array, dimension ( LDA , max(NN) ). It holds the matrix whose eigenvalues are about to be computed. On exit, A contains the last matrix that was actually used.
  • LDA — INTEGER. The leading dimension of A. It must be at least 1 and at least max( NN ).
  • AP — COMPLEX*16 array, dimension ( max(NN)*max(NN+1)/2 ). The matrix A, but stored in packed format. Same queen, smaller closet 🎀
  • SD — DOUBLE PRECISION array, dimension ( max(NN) ). The diagonal of the tridiagonal matrix computed by ZHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A.
  • SE — DOUBLE PRECISION array, dimension ( max(NN) ). The off-diagonal of the tridiagonal matrix computed by ZHETRD. On exit, SD and SE contain the tridiagonal form of the matrix in A. (Yes, same sentence as SD. They're a package deal, iykyk.)
  • D1 — DOUBLE PRECISION array, dimension ( max(NN) ). The eigenvalues of A, as computed by ZSTEQR simultaneously with Z. On exit, the eigenvalues in D1 correspond with the matrix in A.
  • D2 — DOUBLE PRECISION array, dimension ( max(NN) ). The eigenvalues of A, as computed by ZSTEQR if Z is not computed. On exit, the eigenvalues in D2 correspond with the matrix in A.
  • D3 — DOUBLE PRECISION array, dimension ( max(NN) ). The eigenvalues of A, as computed by DSTERF. On exit, the eigenvalues in D3 correspond with the matrix in A.
  • D4 — DOUBLE PRECISION array, dimension ( max(NN) ). The eigenvalues of A, as computed by ZPTEQR(V). ZPTEQR factors S as Z4 D4 Z4*. On exit, the eigenvalues in D4 correspond with the matrix in A.
  • D5 — DOUBLE PRECISION array, dimension ( max(NN) ). The eigenvalues of A, as computed by ZPTEQR(N) when Z is not computed. On exit, the eigenvalues in D4 correspond with the matrix in A. (And yep — the doc really does say D4 there, not D5. We're just the messengers, bestie 🙈.)
  • WA1 — DOUBLE PRECISION array, dimension ( max(NN) ). All eigenvalues of A, computed to high absolute accuracy, with different range options — as computed by DSTEBZ.
  • WA2 — DOUBLE PRECISION array, dimension ( max(NN) ). Selected eigenvalues of A, computed to high absolute accuracy, with different range options — as computed by DSTEBZ. Here's the bit: choose random values for IL and IU, then ask for the IL-th through IU-th eigenvalues.
  • WA3 — DOUBLE PRECISION array, dimension ( max(NN) ). Selected eigenvalues of A, computed to high absolute accuracy, with different range options — as computed by DSTEBZ. This time, determine the values VL and VU of the IL-th and IU-th eigenvalues, then ask for all the eigenvalues in that range.
  • WR — DOUBLE PRECISION array, dimension ( max(NN) ). All eigenvalues of A, computed to high absolute accuracy, with different options — as computed by DSTEBZ.
  • U — COMPLEX*16 array, dimension ( LDU, max(NN) ). The unitary matrix computed by ZHETRD + ZUNGTR. 💜
  • LDU — INTEGER. The leading dimension of U, Z, and V — all three of em, one shared glow-up. It must be at least 1 and at least max( NN ).
  • V — COMPLEX*16 array, dimension ( LDU, max(NN) ). The Householder vectors computed by ZHETRD while reducing A to tridiagonal form. The vectors computed with UPLO='U' live in the upper triangle, and the vectors computed with UPLO='L' live in the lower triangle. (As described in ZHETRD, the sub- and superdiagonal are not set to 1, even though the true Householder vector does have a 1 in that position. The routines that use V, like ZUNGTR, set those entries to 1 before using them, and then restore them later. Considerate little icons 🧚)
  • VP — COMPLEX*16 array, dimension ( max(NN)*max(NN+1)/2 ). The matrix V, but stored in packed format.
  • TAU — COMPLEX*16 array, dimension ( max(NN) ). The Householder factors computed by ZHETRD while reducing A to tridiagonal form.
  • Z — COMPLEX*16 array, dimension ( LDU, max(NN) ). The unitary matrix of eigenvectors computed by ZSTEQR, ZPTEQR, and ZSTEIN.
  • WORK — COMPLEX*16 array, dimension ( LWORK ).
  • LWORK — 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 — INTEGER array. Workspace. That's it, that's the whole description 🫡
  • LIWORK — 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.

  • RWORK — DOUBLE PRECISION array. That's all she wrote on this one ✨
  • LRWORK — INTEGER. The number of entries in LRWORK (dimension( ??? ) — yes, the doc itself leaves a whole question mark situation there. We do not know either 💀).
  • RESULT — DOUBLE PRECISION array, dimension (26) — twenty-six, count em. The values computed by the tests described above. These values are currently limited to 1/ulp, to avoid overflow.
  • INFO — hold that thought bestie, its description picks up right where this part ends 💫

INFO is INTEGER — the official vibe check for this whole routine. If it’s 0, then everything ran OK, no cap. 💖

  • -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 ZLATMR, CLATMS, ZHETRD, ZUNGTR, ZSTEQR, DSTERF, or ZUNMC2 returns an error code, the absolute value of it is returned. 😭

Some Local Variables and Parameters: 💅

The original even throws in a lil ---- ----- --------- --- ---------- underline, iconic.

  • 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 .
  • KMODE(j) — The MODE value to be passed to the matrix generator for type .
  • KMAGN(j) — The order of magnitude ( O(1), O(overflow^(1/2) ), O(underflow^(1/2) )

Author 🫶

  • Univ. of Tennessee
  • Univ. of California Berkeley
  • Univ. of Colorado Denver
  • NAG Ltd.

Definition at line 620 of file zchkst2stg.f.

Author ✨ (auto-generated bestie)

Generated automatically by Doxygen for LAPACK from the source code.

okayyy that’s the page, bestie 🫶 — footer tea: source/version is LAPACK Version 3.12.0, file is TESTING/EIG/zchkst2stg.f(3), and the auto-gen date is living in that footer vibe. bye 💅✨

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