lag2(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

💅 lag2 — ok bestie, this is the 2x2 eig situation

so like. the name of this whole page is lag2 (short for "2x2 eig", which is honestly the most iconique abbreviation ever), and it lives in section 3 of the Library Functions Manual. it's a LAPACK thing. buckle up, we're doing linear algebra 💖

✨ SYNOPSIS ✨

🫶 Functions

  • subroutine dlag2 (a, lda, b, ldb, safmin, scale1, scale2, wr1, wr2, wi) — DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow. (double precision version, she's the ✨ premium ✨ one)
  • subroutine slag2 (a, lda, b, ldb, safmin, scale1, scale2, wr1, wr2, wi) — SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow. (single precision twin sister, same energy 🌸)

🦋 Detailed Description

okay so this is the part where the docs just introduce the two subroutines and then go off on each one in full detail. it's giving very much main character energy. scroll down, we got you 💜

👑 Function Documentation

🎀 subroutine dlag2 (double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision safmin, double precision scale1, double precision scale2, double precision wr1, double precision wr2, double precision wi)

DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

⭐ Purpose:

!>
!> DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
!> problem  A - w B, with scaling as necessary to avoid over-/underflow.
!>
!> The scaling factor  results in a modified eigenvalue equation
!>
!>     s A - w B
!>
!> where  s  is a non-negative scaling factor chosen so that  w,  w B,
!> and  s A  do not overflow and, if possible, do not underflow, either.
!>

like. the whole point is she computes eigenvalues, but the extra part is she scales everything first so nothing overflows or underflows on her watch. literally the group project member who does the work 💅

💖 Parameters

  • A — !> A is DOUBLE PRECISION array, dimension (LDA, 2). On entry, the 2 x 2 matrix A. It is assumed that its 1-norm is less than 1/SAFMIN. Entries less than sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
  • LDA — !> LDA is INTEGER. The leading dimension of the array A. LDA >= 2.
  • B — !> B is DOUBLE PRECISION array, dimension (LDB, 2). On entry, the 2 x 2 upper triangular matrix B. It is assumed that the one-norm of B is less than 1/SAFMIN. The diagonals should be at least sqrt(SAFMIN) times the largest element of B (in absolute value); if a diagonal is smaller than that, then +/- sqrt(SAFMIN) will be used instead of that diagonal.
  • LDB — !> LDB is INTEGER. The leading dimension of the array B. LDB >= 2.
  • SAFMIN — !> SAFMIN is DOUBLE PRECISION. The smallest positive number s.t. 1/SAFMIN does not overflow. (This should always be DLAMCH('S') -- it is an argument in order to avoid having to call DLAMCH frequently.)
  • SCALE1 — !> SCALE1 is DOUBLE PRECISION. A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the first eigenvalue. If the eigenvalues are complex, then the eigenvalues are ( WR1 +/- WI i ) / SCALE1 (which may lie outside the exponent range of the machine), SCALE1=SCALE2, and SCALE1 will always be positive. If the eigenvalues are real, then the first (real) eigenvalue is WR1 / SCALE1 , but this may overflow or underflow, and in fact, SCALE1 may be zero or less than the underflow threshold if the exact eigenvalue is sufficiently large.
  • SCALE2 — !> SCALE2 is DOUBLE PRECISION. A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the second eigenvalue. If the eigenvalues are complex, then SCALE2=SCALE1. If the eigenvalues are real, then the second (real) eigenvalue is WR2 / SCALE2 , but this may overflow or underflow, and in fact, SCALE2 may be zero or less than the underflow threshold if the exact eigenvalue is sufficiently large.
  • WR1 — !> WR1 is DOUBLE PRECISION. If the eigenvalue is real, then WR1 is SCALE1 times the eigenvalue closest to the (2,2) element of A B**(-1). If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues.
  • WR2 — !> WR2 is DOUBLE PRECISION. If the eigenvalue is real, then WR2 is SCALE2 times the other eigenvalue. If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues.
  • WI — !> WI is DOUBLE PRECISION. If the eigenvalue is real, then WI is zero. If the eigenvalue is complex, then WI is SCALE1 times the imaginary part of the eigenvalues. WI will always be non-negative.
🫶 real talk: SCALE1 / SCALE2 can genuinely be zero or fall below the underflow threshold when the exact eigenvalue is big enough. that's not a bug, that's just how the math goes, bestie. don't come for her.

🌈 Author

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

Definition at line 154 of file dlag2.f.

🧚 subroutine slag2 (real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real safmin, real scale1, real scale2, real wr1, real wr2, real wi)

SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

⭐ Purpose:

!>
!> SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
!> problem  A - w B, with scaling as necessary to avoid over-/underflow.
!>
!> The scaling factor  results in a modified eigenvalue equation
!>
!>     s A - w B
!>
!> where  s  is a non-negative scaling factor chosen so that  w,  w B,
!> and  s A  do not overflow and, if possible, do not underflow, either.
!>

same purpose, same s A - w B flex, just in REAL instead of DOUBLE PRECISION. she ate and left no crumbs ✨

💖 Parameters

  • A — !> A is REAL array, dimension (LDA, 2). On entry, the 2 x 2 matrix A. It is assumed that its 1-norm is less than 1/SAFMIN. Entries less than sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
  • LDA — !> LDA is INTEGER. The leading dimension of the array A. LDA >= 2.
  • B — !> B is REAL array, dimension (LDB, 2). On entry, the 2 x 2 upper triangular matrix B. It is assumed that the one-norm of B is less than 1/SAFMIN. The diagonals should be at least sqrt(SAFMIN) times the largest element of B (in absolute value); if a diagonal is smaller than that, then +/- sqrt(SAFMIN) will be used instead of that diagonal.
  • LDB — !> LDB is INTEGER. The leading dimension of the array B. LDB >= 2.
  • SAFMIN — !> SAFMIN is REAL. The smallest positive number s.t. 1/SAFMIN does not overflow. (This should always be SLAMCH('S') -- it is an argument in order to avoid having to call SLAMCH frequently.)
  • SCALE1 — !> SCALE1 is REAL. A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the first eigenvalue. If the eigenvalues are complex, then the eigenvalues are ( WR1 +/- WI i ) / SCALE1 (which may lie outside the exponent range of the machine), SCALE1=SCALE2, and SCALE1 will always be positive. If the eigenvalues are real, then the first (real) eigenvalue is WR1 / SCALE1 , but this may overflow or underflow, and in fact, SCALE1 may be zero or less than the underflow threshold if the exact eigenvalue is sufficiently large.
  • SCALE2 — !> SCALE2 is REAL. A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the second eigenvalue. If the eigenvalues are complex, then SCALE2=SCALE1. If the eigenvalues are real, then the second (real) eigenvalue is WR2 / SCALE2 , but this may overflow or underflow, and in fact, SCALE2 may be zero or less than the underflow threshold if the exact eigenvalue is sufficiently large.
  • WR1 — !> WR1 is REAL. If the eigenvalue is real, then WR1 is SCALE1 times the eigenvalue closest to the (2,2) element of A B**(-1). If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues.
  • WR2 — !> WR2 is REAL. If the eigenvalue is real, then WR2 is SCALE2 times the other eigenvalue. If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues.
  • WI — !> WI is REAL. If the eigenvalue is real, then WI is zero. If the eigenvalue is complex, then WI is SCALE1 times the imaginary part of the eigenvalues. WI will always be non-negative.
💀 literally same caveat as her double-precision sister: the scales can hit zero or dip under the underflow threshold. it's giving "feature, not bug."

🌈 Author

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

Definition at line 154 of file slag2.f.

🌸 Author

Generated automatically by Doxygen for LAPACK from the source code. (so nobody hand-wrote this bad boy, the robots did it 🫡)

and that's the whole page bestie!! LAPACK, Version 3.12.0 — that's the version and the date-stamp vibe, straight from the footer. mwah, go compute some eigenvalues, ily 💅💖

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