PCA_INTERNAL.lambda

=LAMBDA( system, tolerance, LET
  ( nCols,        COLUMNS(system)
  , covMatrix,    INDEX(system, SEQUENCE(nCols), SEQUENCE(1, nCols))
  , accJacobi,    INDEX(system, SEQUENCE(nCols, 1, nCols + 1), SEQUENCE(1, nCols))
  , covLower,     MAKEARRAY(nCols, nCols, LAMBDA(iRow, iCol, IF(iCol > iRow - 1, 0, INDEX(covMatrix, iRow, iCol))))
  , covAbs,       ABS(covLower)
  , worst,        MAX(covAbs)
  , IF( worst < tolerance, system, LET
    ( worstInRow, BYROW(covAbs, LAMBDA(x, MAX(x)))
    , wRow,       MATCH(worst, worstInRow, 0)
    , worstInCol, BYCOL(covAbs, LAMBDA(x, MAX(x)))
    , wCol,       MATCH(worst, worstInCol, 0)
    , tau,        (INDEX(covMatrix, wRow, wRow) - INDEX(covMatrix, wCol, wCol)) / (2 * INDEX(covMatrix, wRow, wCol))
    , t_,         IF(tau < 0, -tau - SQRT(1 + tau*tau), -tau + SQRT(1 + tau*tau))
    , cval,       1 / SQRT(1 + t_ * t_)
    , sval,       cval * t_
    , jacobi,     MAKEARRAY( nCols, nCols, LAMBDA(jRow, jCol, 
      IF(AND(jRow = wCol, jCol = wCol),  cval,
        IF(AND(jRow = wRow, jCol = wRow),  cval,
          IF(AND(jRow = wCol, jCol = wRow),  sval,
            IF(AND(jRow = wRow, jCol = wCol), -sval,
              IF(AND(jRow = jCol), 1, 0)))))))
    , newCovMatrix, MMULT(MMULT(TRANSPOSE(jacobi), covMatrix), jacobi)
    , newAccJacobi, MMULT(TRANSPOSE(jacobi), accJacobi)
    , PCA_INTERNAL(VSTACK(newCovMatrix, newAccJacobi), tolerance)
))))