DCHKST(3)

🦋 dchkst.f(3) — Library Functions Manual 🦋

okay so bestie, this is the Library Functions Manual page for DCHKST, and we're starting from the very top. Buckle up, it's giving Fortran documentation with a side of main character energy. 💅

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✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/TESTING/EIG/dchkst.f

That's the whole NAME section, bestie. Literally just the path. No cap. 🎀

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🌸 SYNOPSIS 🌸

🎀 Functions/Subroutines 🎀

subroutine DCHKST (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, iwork, liwork, result, info)
    DCHKST

That's the subroutine calling card. All the argument names are right there, slaying in order. ✨

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🌈 Function/Subroutine Documentation 🌈

💜 subroutine DCHKST 💜

subroutine DCHKST (integer nsizes, integer, dimension( * ) nn, integer
    ntypes, logical, dimension( * ) dotype, integer, dimension( 4 ) iseed,
    double precision thresh, integer nounit, double precision, dimension(
    lda, * ) a, integer lda, double precision, 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, double
    precision, dimension( ldu, * ) u, integer ldu, double precision,
    dimension( ldu, * ) v, double precision, dimension( * ) vp, double
    precision, dimension( * ) tau, double precision, dimension( ldu, * ) z,
    double precision, dimension( * ) work, integer lwork, integer,
    dimension( * ) iwork, integer liwork, double precision, dimension( * )
    result, integer info)
    DCHKST

Here's the full typed signature, with all the integer, logical, and double precision details intact. Do not lose a single comma or dimension, bestie — this is the exact shape of the subroutine. 🫶

💭 Purpose 💭

the purpose prose is incoming in the next part, bestie — this chunk stops right at the label.

okay so bestie, this is the part where we get into what this thing actually does 💅 buckle up.

DCHKST checks the symmetric eigenvalue problem routines. that's the whole vibe.

• 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; DCHKST 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. iconic behavior tbh.
• DSPTRD does the same as DSYTRD, except that A and V are stored in ____ format.
• DORGTR constructs the matrix U from the contents of V and TAU. king shit 👑
• 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.

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when DCHKST is called, a number of matrix ____ (____) and a number of matrix ____ are specified. (yeah the source has a couple words missing there — not me, i promise 🙈) for each size (____) 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', ... )
4. | I - UV' | / ( n ulp ) — DORGTR( UPLO='L', ... )
5. tests 5-8: same as 1-4, but for DSPTRD and DOPGTR. (yes that's four tests bundled up, slay a little consolidation ✨)
6. | S - Z D Z' | / ( |S| n ulp ) — DSTEQR('V',...)
7. | I - ZZ' | / ( n ulp ) — DSTEQR('V',...)
8. | D1 - D2 | / ( |D1| ulp ) — DSTEQR('N',...)
9. | D1 - D3 | / ( |D1| ulp ) — DSTERF
10. 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, we got more:

11. | S - Z4 D4 Z4' | / ( |S| n ulp ) — DPTEQR('V',...)
12. | I - Z4 Z4' | / ( n ulp ) — DPTEQR('V',...)
13. | D4 - D5 | / ( 100 |D4| ulp ) — DPTEQR('N',...)

and when S is also diagonally dominant by the factor gamma < 1:

14. this whole situation:

         max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,
          i
        omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                            DSTEBZ( 'A', 'E', ...)

15. | WA1 - D3 | / ( |D3| ulp ) — DSTEBZ( 'A', 'E', ...)
16. this double max-min masterpiece:

        ( max { min | WA2(i)-WA3(j) | } +
          i     j
         max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
          i     j
                                            DSTEBZ( 'I', 'E', ...)

17. | S - Y WA1 Y' | / ( |S| n ulp ) — DSTEBZ, SSTEIN
18. | I - Y Y' | / ( n ulp ) — DSTEBZ, SSTEIN
19. | S - Z D Z' | / ( |S| n ulp ) — DSTEDC('I')
20. | I - ZZ' | / ( n ulp ) — DSTEDC('I')
21. | S - Z D Z' | / ( |S| n ulp ) — DSTEDC('V')
22. | I - ZZ' | / ( n ulp ) — DSTEDC('V')
23. | D1 - D2 | / ( |D1| ulp ) — DSTEDC('V') and DSTEDC('N')

okay heads up: test 27 is disabled at the moment because DSTEMR does not guarantee high relative accuracy. so it's sitting there, off, vibing 💀

24. (this is test 27, disabled, remember)

         max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
          i
        omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                            DSTEMR('V', 'A')

25. (test 28, also in the disabled pile)

         max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
          i
        omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
                                            DSTEMR('V', 'I')

and tests 29 through 34 are disabled at present because DSTEMR does not handle partial spectrum requests. so the next few are ghost tests 👻 still documenting them tho, no cap.

26. | S - Z D Z' | / ( |S| n ulp ) — DSTEMR('V', 'I')
27. | I - ZZ' | / ( n ulp ) — DSTEMR('V', 'I')

28.

        ( 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')

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

31.

        ( 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')

32. | S - Z D Z' | / ( |S| n ulp ) — DSTEMR('V', 'A')
33. | I - ZZ' | / ( n ulp ) — DSTEMR('V', 'A')

34.

        ( 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')

and that's the full test roster, bestie. 37-ish whole personality traits 🎀

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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:

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 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 ____ 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.

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💖 Parameters 💖

NSIZES

NSIZES is INTEGER

The number of sizes of matrices to use. If it is zero, DCHKST does nothing. It must be at least zero. (zero = it simply does not perceive you 😭)

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, DCHKST 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.. iykyk 🫶

DOTYPE

…and for DOTYPE, bestie: it’s a 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 DCHKST to continue the same random number sequence. 🎲
• THRESH — THRESH is DOUBLE PRECISION. A test will count as if the , 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
• 0 — everything ran OK. we love a clean run 💖
• -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. bestie pls give it more room 😭

and ok so if DLATMR, SLATMS, DSYTRD, DORGTR, DSTEQR, SSTERF, or DORMC2 returns an error code, the absolute value of it is what comes back at you. no cap, that's the deal 👑

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✨ Some Local Variables and Parameters ✨

okay this is the little cast list of the internals, and honestly it's giving backstage tour energy 🎀

• ZERO, ONE — Real 0 and 1. iconic, no notes.
• 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. (we don't talk about these ones 🙈)
• 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.

and then the following four arrays decode JTYPE, lowkey the cheat sheet of the whole thing 🧚

• 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) ) )

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💅 Author

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

definition at line 587 of file dchkst.f. look at her, line 587, thriving ✨

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👑 Author

generated automatically by Doxygen for LAPACK from the source code. yes bestie, the robots wrote this one 🤖💜

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and that's the whole page, we made it to the end 🫶 this was LAPACK, Version 3.12.0, rendered from /home/abui...IG/dchkst.f(3) — mwah, go forth and check those tridiagonal eigenvalues, you absolute icon 🌈💖