lasq2(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

✨ lasq2 ✨

okay so this one is from the Library Functions Manual, and it's literally the page for lasq2 — aka the dqds step 🎀 (yes that's the whole NAME, short and iconic, we love a minimal queen).

🌸 SYNOPSIS 🌸

💫 Functions 💫

  • subroutine dlasq2 (n, z, info) — DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.
  • subroutine slasq2 (n, z, info) — SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.

(bestie they are the same function, one is just the double precision version and one is the real version. twins fr fr 👯)

🦋 Detailed Description 🦋

like, the whole point of this page is the function documentation, so let's get into it ⬇️

🔥 Function Documentation 🔥

🧚 subroutine dlasq2 (integer n, double precision, dimension( * ) z, integer info)

DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.

🌈 Purpose 🌈

DLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd array Z to high relative accuracy are computed to high relative accuracy, in the absence of denormalization, underflow and overflow.

To see the relation of Z to the tridiagonal matrix, let L be a unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and let U be an upper bidiagonal matrix with 1's above and diagonal Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the symmetric tridiagonal to which it is similar.

💀 Note: DLASQ2 defines a logical variable, IEEE, which is true on machines which follow ieee-754 floating-point standard in their handling of infinities and NaNs, and false otherwise. This variable is passed to DLASQ3.

🍓 Parameters 🍓

  • N — N is INTEGER. The number of rows and columns in the matrix. N >= 0.
  • Z — Z is DOUBLE PRECISION array, dimension ( 4*N ). On entry Z holds the qd array. On exit, entries 1 to N hold the eigenvalues in decreasing order, Z( 2*N+1 ) holds the trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 ) holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of shifts that failed. 🫶
  • INFO — INFO is INTEGER:
    • = 0: successful exit 🎉
    • < 0: if the i-th argument is a scalar and had an illegal value, then INFO = -i; if the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j)
    • > 0: the algorithm failed 😭
      • = 1, a split was marked by a positive value in E
      • = 2, current block of Z not diagonalized after 100*N iterations (in inner while loop). On exit Z holds a qd array with the same eigenvalues as the given Z.
      • = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)

👑 Author 👑

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

💜 Further Details 💜

Local Variables: I0:N0 defines a current unreduced segment of Z. The shifts are accumulated in SIGMA. Iteration count is in ITER. Ping-pong is controlled by PP (alternates between 0 and 1). 🏓

Definition at line 111 of file dlasq2.f.

🧁 subroutine slasq2 (integer n, real, dimension( * ) z, integer info)

SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd Array Z to high relative accuracy. Used by sbdsqr and sstegr.

🌈 Purpose 🌈

SLASQ2 computes all the eigenvalues of the symmetric positive definite tridiagonal matrix associated with the qd array Z to high relative accuracy are computed to high relative accuracy, in the absence of denormalization, underflow and overflow.

To see the relation of Z to the tridiagonal matrix, let L be a unit lower bidiagonal matrix with subdiagonals Z(2,4,6,,..) and let U be an upper bidiagonal matrix with 1's above and diagonal Z(1,3,5,,..). The tridiagonal is L*U or, if you prefer, the symmetric tridiagonal to which it is similar.

💀 Note: SLASQ2 defines a logical variable, IEEE, which is true on machines which follow ieee-754 floating-point standard in their handling of infinities and NaNs, and false otherwise. This variable is passed to SLASQ3.

🍓 Parameters 🍓

  • N — N is INTEGER. The number of rows and columns in the matrix. N >= 0.
  • Z — Z is REAL array, dimension ( 4*N ). On entry Z holds the qd array. On exit, entries 1 to N hold the eigenvalues in decreasing order, Z( 2*N+1 ) holds the trace, and Z( 2*N+2 ) holds the sum of the eigenvalues. If N > 2, then Z( 2*N+3 ) holds the iteration count, Z( 2*N+4 ) holds NDIVS/NIN^2, and Z( 2*N+5 ) holds the percentage of shifts that failed. 🫶
  • INFO — INFO is INTEGER:
    • = 0: successful exit 🎉
    • < 0: if the i-th argument is a scalar and had an illegal value, then INFO = -i; if the i-th argument is an array and the j-entry had an illegal value, then INFO = -(i*100+j)
    • > 0: the algorithm failed 😭
      • = 1, a split was marked by a positive value in E
      • = 2, current block of Z not diagonalized after 100*N iterations (in inner while loop). On exit Z holds a qd array with the same eigenvalues as the given Z.
      • = 3, termination criterion of outer while loop not met (program created more than N unreduced blocks)

👑 Author 👑

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

💜 Further Details 💜

Local Variables: I0:N0 defines a current unreduced segment of Z. The shifts are accumulated in SIGMA. Iteration count is in ITER. Ping-pong is controlled by PP (alternates between 0 and 1). 🏓

Definition at line 111 of file slasq2.f.

🫶 Author 🫶

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okay that's the whole page bestie, and the tea at the very bottom: this is LAPACK, Version 3.12.0 🎀✨ — that's the source/version line, consider it your cute sign-off. mwah, go diagonalize something 💖

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