test (#52)

[bytom/vapor.git] / vendor / gonum.org / v1 / gonum / lapack / gonum / dlaln2.go

// Copyright ©2016 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package gonum

import "math"

// Dlaln2 solves a linear equation or a system of 2 linear equations of the form
//  (ca A   - w D) X = scale B,  if trans == false,
//  (ca A^T - w D) X = scale B,  if trans == true,
// where A is a na×na real matrix, ca is a real scalar, D is a na×na diagonal
// real matrix, w is a scalar, real if nw == 1, complex if nw == 2, and X and B
// are na×1 matrices, real if w is real, complex if w is complex.
//
// If w is complex, X and B are represented as na×2 matrices, the first column
// of each being the real part and the second being the imaginary part.
//
// na and nw must be 1 or 2, otherwise Dlaln2 will panic.
//
// d1 and d2 are the diagonal elements of D. d2 is not used if na == 1.
//
// wr and wi represent the real and imaginary part, respectively, of the scalar
// w. wi is not used if nw == 1.
//
// smin is the desired lower bound on the singular values of A. This should be
// a safe distance away from underflow or overflow, say, between
// (underflow/machine precision) and (overflow*machine precision).
//
// If both singular values of (ca A - w D) are less than smin, smin*identity
// will be used instead of (ca A - w D). If only one singular value is less than
// smin, one element of (ca A - w D) will be perturbed enough to make the
// smallest singular value roughly smin. If both singular values are at least
// smin, (ca A - w D) will not be perturbed. In any case, the perturbation will
// be at most some small multiple of max(smin, ulp*norm(ca A - w D)). The
// singular values are computed by infinity-norm approximations, and thus will
// only be correct to a factor of 2 or so.
//
// All input quantities are assumed to be smaller than overflow by a reasonable
// factor.
//
// scale is a scaling factor less than or equal to 1 which is chosen so that X
// can be computed without overflow. X is further scaled if necessary to assure
// that norm(ca A - w D)*norm(X) is less than overflow.
//
// xnorm contains the infinity-norm of X when X is regarded as a na×nw real
// matrix.
//
// ok will be false if (ca A - w D) had to be perturbed to make its smallest
// singular value greater than smin, otherwise ok will be true.
//
// Dlaln2 is an internal routine. It is exported for testing purposes.
func (impl Implementation) Dlaln2(trans bool, na, nw int, smin, ca float64, a []float64, lda int, d1, d2 float64, b []float64, ldb int, wr, wi float64, x []float64, ldx int) (scale, xnorm float64, ok bool) {
        // TODO(vladimir-ch): Consider splitting this function into two, one
        // handling the real case (nw == 1) and the other handling the complex
        // case (nw == 2). Given that Go has complex types, their signatures
        // would be simpler and more natural, and the implementation not as
        // convoluted.

        if na != 1 && na != 2 {
                panic("lapack: invalid value of na")
        }
        if nw != 1 && nw != 2 {
                panic("lapack: invalid value of nw")
        }
        checkMatrix(na, na, a, lda)
        checkMatrix(na, nw, b, ldb)
        checkMatrix(na, nw, x, ldx)

        smlnum := 2 * dlamchS
        bignum := 1 / smlnum
        smini := math.Max(smin, smlnum)

        ok = true
        scale = 1

        if na == 1 {
                // 1×1 (i.e., scalar) system C X = B.

                if nw == 1 {
                        // Real 1×1 system.

                        // C = ca A - w D.
                        csr := ca*a[0] - wr*d1
                        cnorm := math.Abs(csr)

                        // If |C| < smini, use C = smini.
                        if cnorm < smini {
                                csr = smini
                                cnorm = smini
                                ok = false
                        }

                        // Check scaling for X = B / C.
                        bnorm := math.Abs(b[0])
                        if cnorm < 1 && bnorm > math.Max(1, bignum*cnorm) {
                                scale = 1 / bnorm
                        }

                        // Compute X.
                        x[0] = b[0] * scale / csr
                        xnorm = math.Abs(x[0])

                        return scale, xnorm, ok
                }

                // Complex 1×1 system (w is complex).

                // C = ca A - w D.
                csr := ca*a[0] - wr*d1
                csi := -wi * d1
                cnorm := math.Abs(csr) + math.Abs(csi)

                // If |C| < smini, use C = smini.
                if cnorm < smini {
                        csr = smini
                        csi = 0
                        cnorm = smini
                        ok = false
                }

                // Check scaling for X = B / C.
                bnorm := math.Abs(b[0]) + math.Abs(b[1])
                if cnorm < 1 && bnorm > math.Max(1, bignum*cnorm) {
                        scale = 1 / bnorm
                }

                // Compute X.
                cx := complex(scale*b[0], scale*b[1]) / complex(csr, csi)
                x[0], x[1] = real(cx), imag(cx)
                xnorm = math.Abs(x[0]) + math.Abs(x[1])

                return scale, xnorm, ok
        }

        // 2×2 system.

        // Compute the real part of
        //  C = ca A   - w D
        // or
        //  C = ca A^T - w D.
        crv := [4]float64{
                ca*a[0] - wr*d1,
                ca * a[1],
                ca * a[lda],
                ca*a[lda+1] - wr*d2,
        }
        if trans {
                crv[1] = ca * a[lda]
                crv[2] = ca * a[1]
        }

        pivot := [4][4]int{
                {0, 1, 2, 3},
                {1, 0, 3, 2},
                {2, 3, 0, 1},
                {3, 2, 1, 0},
        }

        if nw == 1 {
                // Real 2×2 system (w is real).

                // Find the largest element in C.
                var cmax float64
                var icmax int
                for j, v := range crv {
                        v = math.Abs(v)
                        if v > cmax {
                                cmax = v
                                icmax = j
                        }
                }

                // If norm(C) < smini, use smini*identity.
                if cmax < smini {
                        bnorm := math.Max(math.Abs(b[0]), math.Abs(b[ldb]))
                        if smini < 1 && bnorm > math.Max(1, bignum*smini) {
                                scale = 1 / bnorm
                        }
                        temp := scale / smini
                        x[0] = temp * b[0]
                        x[ldx] = temp * b[ldb]
                        xnorm = temp * bnorm
                        ok = false

                        return scale, xnorm, ok
                }

                // Gaussian elimination with complete pivoting.
                // Form upper triangular matrix
                //  [ur11 ur12]
                //  [   0 ur22]
                ur11 := crv[icmax]
                ur12 := crv[pivot[icmax][1]]
                cr21 := crv[pivot[icmax][2]]
                cr22 := crv[pivot[icmax][3]]
                ur11r := 1 / ur11
                lr21 := ur11r * cr21
                ur22 := cr22 - ur12*lr21

                // If smaller pivot < smini, use smini.
                if math.Abs(ur22) < smini {
                        ur22 = smini
                        ok = false
                }

                var br1, br2 float64
                if icmax > 1 {
                        // If the pivot lies in the second row, swap the rows.
                        br1 = b[ldb]
                        br2 = b[0]
                } else {
                        br1 = b[0]
                        br2 = b[ldb]
                }
                br2 -= lr21 * br1 // Apply the Gaussian elimination step to the right-hand side.

                bbnd := math.Max(math.Abs(ur22*ur11r*br1), math.Abs(br2))
                if bbnd > 1 && math.Abs(ur22) < 1 && bbnd >= bignum*math.Abs(ur22) {
                        scale = 1 / bbnd
                }

                // Solve the linear system ur*xr=br.
                xr2 := br2 * scale / ur22
                xr1 := scale*br1*ur11r - ur11r*ur12*xr2
                if icmax&0x1 != 0 {
                        // If the pivot lies in the second column, swap the components of the solution.
                        x[0] = xr2
                        x[ldx] = xr1
                } else {
                        x[0] = xr1
                        x[ldx] = xr2
                }
                xnorm = math.Max(math.Abs(xr1), math.Abs(xr2))

                // Further scaling if norm(A)*norm(X) > overflow.
                if xnorm > 1 && cmax > 1 && xnorm > bignum/cmax {
                        temp := cmax / bignum
                        x[0] *= temp
                        x[ldx] *= temp
                        xnorm *= temp
                        scale *= temp
                }

                return scale, xnorm, ok
        }

        // Complex 2×2 system (w is complex).

        // Find the largest element in C.
        civ := [4]float64{
                -wi * d1,
                0,
                0,
                -wi * d2,
        }
        var cmax float64
        var icmax int
        for j, v := range crv {
                v := math.Abs(v)
                if v+math.Abs(civ[j]) > cmax {
                        cmax = v + math.Abs(civ[j])
                        icmax = j
                }
        }

        // If norm(C) < smini, use smini*identity.
        if cmax < smini {
                br1 := math.Abs(b[0]) + math.Abs(b[1])
                br2 := math.Abs(b[ldb]) + math.Abs(b[ldb+1])
                bnorm := math.Max(br1, br2)
                if smini < 1 && bnorm > 1 && bnorm > bignum*smini {
                        scale = 1 / bnorm
                }
                temp := scale / smini
                x[0] = temp * b[0]
                x[1] = temp * b[1]
                x[ldb] = temp * b[ldb]
                x[ldb+1] = temp * b[ldb+1]
                xnorm = temp * bnorm
                ok = false

                return scale, xnorm, ok
        }

        // Gaussian elimination with complete pivoting.
        ur11 := crv[icmax]
        ui11 := civ[icmax]
        ur12 := crv[pivot[icmax][1]]
        ui12 := civ[pivot[icmax][1]]
        cr21 := crv[pivot[icmax][2]]
        ci21 := civ[pivot[icmax][2]]
        cr22 := crv[pivot[icmax][3]]
        ci22 := civ[pivot[icmax][3]]
        var (
                ur11r, ui11r float64
                lr21, li21   float64
                ur12s, ui12s float64
                ur22, ui22   float64
        )
        if icmax == 0 || icmax == 3 {
                // Off-diagonals of pivoted C are real.
                if math.Abs(ur11) > math.Abs(ui11) {
                        temp := ui11 / ur11
                        ur11r = 1 / (ur11 * (1 + temp*temp))
                        ui11r = -temp * ur11r
                } else {
                        temp := ur11 / ui11
                        ui11r = -1 / (ui11 * (1 + temp*temp))
                        ur11r = -temp * ui11r
                }
                lr21 = cr21 * ur11r
                li21 = cr21 * ui11r
                ur12s = ur12 * ur11r
                ui12s = ur12 * ui11r
                ur22 = cr22 - ur12*lr21
                ui22 = ci22 - ur12*li21
        } else {
                // Diagonals of pivoted C are real.
                ur11r = 1 / ur11
                // ui11r is already 0.
                lr21 = cr21 * ur11r
                li21 = ci21 * ur11r
                ur12s = ur12 * ur11r
                ui12s = ui12 * ur11r
                ur22 = cr22 - ur12*lr21 + ui12*li21
                ui22 = -ur12*li21 - ui12*lr21
        }
        u22abs := math.Abs(ur22) + math.Abs(ui22)

        // If smaller pivot < smini, use smini.
        if u22abs < smini {
                ur22 = smini
                ui22 = 0
                ok = false
        }

        var br1, bi1 float64
        var br2, bi2 float64
        if icmax > 1 {
                // If the pivot lies in the second row, swap the rows.
                br1 = b[ldb]
                bi1 = b[ldb+1]
                br2 = b[0]
                bi2 = b[1]
        } else {
                br1 = b[0]
                bi1 = b[1]
                br2 = b[ldb]
                bi2 = b[ldb+1]
        }
        br2 += -lr21*br1 + li21*bi1
        bi2 += -li21*br1 - lr21*bi1

        bbnd1 := u22abs * (math.Abs(ur11r) + math.Abs(ui11r)) * (math.Abs(br1) + math.Abs(bi1))
        bbnd2 := math.Abs(br2) + math.Abs(bi2)
        bbnd := math.Max(bbnd1, bbnd2)
        if bbnd > 1 && u22abs < 1 && bbnd >= bignum*u22abs {
                scale = 1 / bbnd
                br1 *= scale
                bi1 *= scale
                br2 *= scale
                bi2 *= scale
        }

        cx2 := complex(br2, bi2) / complex(ur22, ui22)
        xr2, xi2 := real(cx2), imag(cx2)
        xr1 := ur11r*br1 - ui11r*bi1 - ur12s*xr2 + ui12s*xi2
        xi1 := ui11r*br1 + ur11r*bi1 - ui12s*xr2 - ur12s*xi2
        if icmax&0x1 != 0 {
                // If the pivot lies in the second column, swap the components of the solution.
                x[0] = xr2
                x[1] = xi2
                x[ldx] = xr1
                x[ldx+1] = xi1
        } else {
                x[0] = xr1
                x[1] = xi1
                x[ldx] = xr2
                x[ldx+1] = xi2
        }
        xnorm = math.Max(math.Abs(xr1)+math.Abs(xi1), math.Abs(xr2)+math.Abs(xi2))

        // Further scaling if norm(A)*norm(X) > overflow.
        if xnorm > 1 && cmax > 1 && xnorm > bignum/cmax {
                temp := cmax / bignum
                x[0] *= temp
                x[1] *= temp
                x[ldx] *= temp
                x[ldx+1] *= temp
                xnorm *= temp
                scale *= temp
        }

        return scale, xnorm, ok
}