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656 lines
16 KiB
Go
656 lines
16 KiB
Go
// Copyright ©2017 The gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package testlapack
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import (
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"math"
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"math/rand"
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"gonum.org/v1/gonum/blas"
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"gonum.org/v1/gonum/blas/blas64"
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)
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// Dlatm1 computes the entries of dst as specified by mode, cond and rsign.
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//
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// mode describes how dst will be computed:
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// |mode| == 1: dst[0] = 1 and dst[1:n] = 1/cond
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// |mode| == 2: dst[:n-1] = 1/cond and dst[n-1] = 1
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// |mode| == 3: dst[i] = cond^{-i/(n-1)}, i=0,...,n-1
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// |mode| == 4: dst[i] = 1 - i*(1-1/cond)/(n-1)
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// |mode| == 5: dst[i] = random number in the range (1/cond, 1) such that
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// their logarithms are uniformly distributed
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// |mode| == 6: dst[i] = random number from the distribution given by dist
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// If mode is negative, the order of the elements of dst will be reversed.
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// For other values of mode Dlatm1 will panic.
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//
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// If rsign is true and mode is not ±6, each entry of dst will be multiplied by 1
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// or -1 with probability 0.5
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//
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// dist specifies the type of distribution to be used when mode == ±6:
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// dist == 1: Uniform[0,1)
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// dist == 2: Uniform[-1,1)
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// dist == 3: Normal(0,1)
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// For other values of dist Dlatm1 will panic.
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//
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// rnd is used as a source of random numbers.
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func Dlatm1(dst []float64, mode int, cond float64, rsign bool, dist int, rnd *rand.Rand) {
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amode := mode
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if amode < 0 {
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amode = -amode
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}
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if amode < 1 || 6 < amode {
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panic("testlapack: invalid mode")
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}
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if cond < 1 {
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panic("testlapack: cond < 1")
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}
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if amode == 6 && (dist < 1 || 3 < dist) {
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panic("testlapack: invalid dist")
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}
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n := len(dst)
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if n == 0 {
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return
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}
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switch amode {
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case 1:
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dst[0] = 1
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for i := 1; i < n; i++ {
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dst[i] = 1 / cond
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}
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case 2:
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for i := 0; i < n-1; i++ {
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dst[i] = 1
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}
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dst[n-1] = 1 / cond
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case 3:
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dst[0] = 1
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if n > 1 {
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alpha := math.Pow(cond, -1/float64(n-1))
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for i := 1; i < n; i++ {
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dst[i] = math.Pow(alpha, float64(i))
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}
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}
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case 4:
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dst[0] = 1
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if n > 1 {
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condInv := 1 / cond
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alpha := (1 - condInv) / float64(n-1)
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for i := 1; i < n; i++ {
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dst[i] = float64(n-i-1)*alpha + condInv
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}
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}
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case 5:
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alpha := math.Log(1 / cond)
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for i := range dst {
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dst[i] = math.Exp(alpha * rnd.Float64())
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}
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case 6:
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switch dist {
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case 1:
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for i := range dst {
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dst[i] = rnd.Float64()
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}
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case 2:
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for i := range dst {
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dst[i] = 2*rnd.Float64() - 1
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}
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case 3:
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for i := range dst {
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dst[i] = rnd.NormFloat64()
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}
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}
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}
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if rsign && amode != 6 {
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for i, v := range dst {
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if rnd.Float64() < 0.5 {
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dst[i] = -v
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}
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}
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}
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if mode < 0 {
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for i := 0; i < n/2; i++ {
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dst[i], dst[n-i-1] = dst[n-i-1], dst[i]
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}
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}
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}
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// Dlagsy generates an n×n symmetric matrix A, by pre- and post- multiplying a
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// real diagonal matrix D with a random orthogonal matrix:
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// A = U * D * U^T.
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//
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// work must have length at least 2*n, otherwise Dlagsy will panic.
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//
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// The parameter k is unused but it must satisfy
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// 0 <= k <= n-1.
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func Dlagsy(n, k int, d []float64, a []float64, lda int, rnd *rand.Rand, work []float64) {
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checkMatrix(n, n, a, lda)
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if k < 0 || max(0, n-1) < k {
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panic("testlapack: invalid value of k")
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}
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if len(d) != n {
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panic("testlapack: bad length of d")
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}
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if len(work) < 2*n {
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panic("testlapack: insufficient work length")
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}
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// Initialize lower triangle of A to diagonal matrix.
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for i := 1; i < n; i++ {
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for j := 0; j < i; j++ {
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a[i*lda+j] = 0
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}
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}
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for i := 0; i < n; i++ {
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a[i*lda+i] = d[i]
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}
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bi := blas64.Implementation()
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// Generate lower triangle of symmetric matrix.
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for i := n - 2; i >= 0; i-- {
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for j := 0; j < n-i; j++ {
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work[j] = rnd.NormFloat64()
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}
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wn := bi.Dnrm2(n-i, work[:n-i], 1)
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wa := math.Copysign(wn, work[0])
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var tau float64
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if wn != 0 {
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wb := work[0] + wa
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bi.Dscal(n-i-1, 1/wb, work[1:n-i], 1)
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work[0] = 1
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tau = wb / wa
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}
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// Apply random reflection to A[i:n,i:n] from the left and the
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// right.
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//
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// Compute y := tau * A * u.
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bi.Dsymv(blas.Lower, n-i, tau, a[i*lda+i:], lda, work[:n-i], 1, 0, work[n:2*n-i], 1)
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// Compute v := y - 1/2 * tau * ( y, u ) * u.
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alpha := -0.5 * tau * bi.Ddot(n-i, work[n:2*n-i], 1, work[:n-i], 1)
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bi.Daxpy(n-i, alpha, work[:n-i], 1, work[n:2*n-i], 1)
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// Apply the transformation as a rank-2 update to A[i:n,i:n].
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bi.Dsyr2(blas.Lower, n-i, -1, work[:n-i], 1, work[n:2*n-i], 1, a[i*lda+i:], lda)
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}
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// Store full symmetric matrix.
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for i := 1; i < n; i++ {
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for j := 0; j < i; j++ {
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a[j*lda+i] = a[i*lda+j]
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}
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}
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}
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// Dlagge generates a real general m×n matrix A, by pre- and post-multiplying
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// a real diagonal matrix D with random orthogonal matrices:
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// A = U*D*V.
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//
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// d must have length min(m,n), and work must have length m+n, otherwise Dlagge
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// will panic.
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//
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// The parameters ku and kl are unused but they must satisfy
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// 0 <= kl <= m-1,
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// 0 <= ku <= n-1.
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func Dlagge(m, n, kl, ku int, d []float64, a []float64, lda int, rnd *rand.Rand, work []float64) {
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checkMatrix(m, n, a, lda)
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if kl < 0 || max(0, m-1) < kl {
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panic("testlapack: invalid value of kl")
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}
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if ku < 0 || max(0, n-1) < ku {
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panic("testlapack: invalid value of ku")
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}
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if len(d) != min(m, n) {
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panic("testlapack: bad length of d")
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}
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if len(work) < m+n {
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panic("testlapack: insufficient work length")
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}
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// Initialize A to diagonal matrix.
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for i := 0; i < m; i++ {
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for j := 0; j < n; j++ {
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a[i*lda+j] = 0
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}
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}
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for i := 0; i < min(m, n); i++ {
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a[i*lda+i] = d[i]
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}
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// Quick exit if the user wants a diagonal matrix.
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// if kl == 0 && ku == 0 {
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// return
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// }
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bi := blas64.Implementation()
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// Pre- and post-multiply A by random orthogonal matrices.
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for i := min(m, n) - 1; i >= 0; i-- {
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if i < m-1 {
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for j := 0; j < m-i; j++ {
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work[j] = rnd.NormFloat64()
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}
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wn := bi.Dnrm2(m-i, work[:m-i], 1)
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wa := math.Copysign(wn, work[0])
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var tau float64
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if wn != 0 {
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wb := work[0] + wa
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bi.Dscal(m-i-1, 1/wb, work[1:m-i], 1)
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work[0] = 1
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tau = wb / wa
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}
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// Multiply A[i:m,i:n] by random reflection from the left.
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bi.Dgemv(blas.Trans, m-i, n-i,
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1, a[i*lda+i:], lda, work[:m-i], 1,
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0, work[m:m+n-i], 1)
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bi.Dger(m-i, n-i,
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-tau, work[:m-i], 1, work[m:m+n-i], 1,
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a[i*lda+i:], lda)
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}
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if i < n-1 {
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for j := 0; j < n-i; j++ {
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work[j] = rnd.NormFloat64()
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}
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wn := bi.Dnrm2(n-i, work[:n-i], 1)
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wa := math.Copysign(wn, work[0])
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var tau float64
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if wn != 0 {
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wb := work[0] + wa
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bi.Dscal(n-i-1, 1/wb, work[1:n-i], 1)
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work[0] = 1
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tau = wb / wa
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}
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// Multiply A[i:m,i:n] by random reflection from the right.
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bi.Dgemv(blas.NoTrans, m-i, n-i,
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1, a[i*lda+i:], lda, work[:n-i], 1,
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0, work[n:n+m-i], 1)
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bi.Dger(m-i, n-i,
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-tau, work[n:n+m-i], 1, work[:n-i], 1,
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a[i*lda+i:], lda)
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}
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}
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// TODO(vladimir-ch): Reduce number of subdiagonals to kl and number of
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// superdiagonals to ku.
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}
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// dlarnv fills dst with random numbers from a uniform or normal distribution
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// specified by dist:
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// dist=1: uniform(0,1),
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// dist=2: uniform(-1,1),
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// dist=3: normal(0,1).
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// For other values of dist dlarnv will panic.
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func dlarnv(dst []float64, dist int, rnd *rand.Rand) {
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switch dist {
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default:
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panic("testlapack: invalid dist")
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case 1:
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for i := range dst {
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dst[i] = rnd.Float64()
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}
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case 2:
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for i := range dst {
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dst[i] = 2*rnd.Float64() - 1
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}
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case 3:
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for i := range dst {
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dst[i] = rnd.NormFloat64()
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}
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}
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}
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// dlattr generates an n×n triangular test matrix A with its properties uniquely
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// determined by imat and uplo, and returns whether A has unit diagonal. If diag
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// is blas.Unit, the diagonal elements are set so that A[k,k]=k.
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//
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// trans specifies whether the matrix A or its transpose will be used.
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//
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// If imat is greater than 10, dlattr also generates the right hand side of the
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// linear system A*x=b, or A^T*x=b. Valid values of imat are 7, and all between 11
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// and 19, inclusive.
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//
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// b mush have length n, and work must have length 3*n, and dlattr will panic
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// otherwise.
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func dlattr(imat int, uplo blas.Uplo, trans blas.Transpose, n int, a []float64, lda int, b, work []float64, rnd *rand.Rand) (diag blas.Diag) {
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checkMatrix(n, n, a, lda)
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if len(b) != n {
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panic("testlapack: bad length of b")
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}
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if len(work) < 3*n {
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panic("testlapack: insufficient length of work")
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}
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if uplo != blas.Upper && uplo != blas.Lower {
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panic("testlapack: bad uplo")
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}
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if trans != blas.Trans && trans != blas.NoTrans {
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panic("testlapack: bad trans")
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}
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if n == 0 {
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return blas.NonUnit
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}
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ulp := dlamchE * dlamchB
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smlnum := dlamchS
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bignum := (1 - ulp) / smlnum
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bi := blas64.Implementation()
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switch imat {
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default:
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// TODO(vladimir-ch): Implement the remaining cases.
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panic("testlapack: invalid or unimplemented imat")
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case 7:
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// Identity matrix. The diagonal is set to NaN.
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diag = blas.Unit
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switch uplo {
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case blas.Upper:
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for i := 0; i < n; i++ {
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a[i*lda+i] = math.NaN()
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for j := i + 1; j < n; j++ {
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a[i*lda+j] = 0
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}
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}
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case blas.Lower:
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for i := 0; i < n; i++ {
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for j := 0; j < i; j++ {
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a[i*lda+j] = 0
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}
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a[i*lda+i] = math.NaN()
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}
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}
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case 11:
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// Generate a triangular matrix with elements between -1 and 1,
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// give the diagonal norm 2 to make it well-conditioned, and
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// make the right hand side large so that it requires scaling.
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diag = blas.NonUnit
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switch uplo {
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case blas.Upper:
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for i := 0; i < n-1; i++ {
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dlarnv(a[i*lda+i:i*lda+n], 2, rnd)
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}
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case blas.Lower:
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for i := 1; i < n; i++ {
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dlarnv(a[i*lda:i*lda+i+1], 2, rnd)
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}
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}
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for i := 0; i < n; i++ {
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a[i*lda+i] = math.Copysign(2, a[i*lda+i])
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}
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// Set the right hand side so that the largest value is bignum.
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dlarnv(b, 2, rnd)
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imax := bi.Idamax(n, b, 1)
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bscal := bignum / math.Max(1, b[imax])
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bi.Dscal(n, bscal, b, 1)
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case 12:
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// Make the first diagonal element in the solve small to cause
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// immediate overflow when dividing by T[j,j]. The off-diagonal
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// elements are small (cnorm[j] < 1).
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diag = blas.NonUnit
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tscal := 1 / math.Max(1, float64(n-1))
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switch uplo {
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case blas.Upper:
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for i := 0; i < n; i++ {
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dlarnv(a[i*lda+i:i*lda+n], 2, rnd)
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bi.Dscal(n-i-1, tscal, a[i*lda+i+1:], 1)
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a[i*lda+i] = math.Copysign(1, a[i*lda+i])
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}
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a[(n-1)*lda+n-1] *= smlnum
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case blas.Lower:
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for i := 0; i < n; i++ {
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dlarnv(a[i*lda:i*lda+i+1], 2, rnd)
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bi.Dscal(i, tscal, a[i*lda:], 1)
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a[i*lda+i] = math.Copysign(1, a[i*lda+i])
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}
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a[0] *= smlnum
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}
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dlarnv(b, 2, rnd)
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case 13:
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// Make the first diagonal element in the solve small to cause
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// immediate overflow when dividing by T[j,j]. The off-diagonal
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// elements are O(1) (cnorm[j] > 1).
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diag = blas.NonUnit
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switch uplo {
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case blas.Upper:
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for i := 0; i < n; i++ {
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dlarnv(a[i*lda+i:i*lda+n], 2, rnd)
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a[i*lda+i] = math.Copysign(1, a[i*lda+i])
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}
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a[(n-1)*lda+n-1] *= smlnum
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case blas.Lower:
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for i := 0; i < n; i++ {
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dlarnv(a[i*lda:i*lda+i+1], 2, rnd)
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a[i*lda+i] = math.Copysign(1, a[i*lda+i])
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}
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a[0] *= smlnum
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}
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dlarnv(b, 2, rnd)
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case 14:
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// T is diagonal with small numbers on the diagonal to
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// make the growth factor underflow, but a small right hand side
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// chosen so that the solution does not overflow.
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diag = blas.NonUnit
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switch uplo {
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case blas.Upper:
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for i := 0; i < n; i++ {
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for j := i + 1; j < n; j++ {
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a[i*lda+j] = 0
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}
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if (n-1-i)&0x2 == 0 {
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a[i*lda+i] = smlnum
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} else {
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a[i*lda+i] = 1
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}
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}
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case blas.Lower:
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for i := 0; i < n; i++ {
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for j := 0; j < i; j++ {
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a[i*lda+j] = 0
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}
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if i&0x2 == 0 {
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a[i*lda+i] = smlnum
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} else {
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a[i*lda+i] = 1
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}
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}
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}
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// Set the right hand side alternately zero and small.
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switch uplo {
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case blas.Upper:
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b[0] = 0
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for i := n - 1; i > 0; i -= 2 {
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b[i] = 0
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b[i-1] = smlnum
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}
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case blas.Lower:
|
||
for i := 0; i < n-1; i += 2 {
|
||
b[i] = 0
|
||
b[i+1] = smlnum
|
||
}
|
||
b[n-1] = 0
|
||
}
|
||
case 15:
|
||
// Make the diagonal elements small to cause gradual overflow
|
||
// when dividing by T[j,j]. To control the amount of scaling
|
||
// needed, the matrix is bidiagonal.
|
||
diag = blas.NonUnit
|
||
texp := 1 / math.Max(1, float64(n-1))
|
||
tscal := math.Pow(smlnum, texp)
|
||
switch uplo {
|
||
case blas.Upper:
|
||
for i := 0; i < n; i++ {
|
||
a[i*lda+i] = tscal
|
||
if i < n-1 {
|
||
a[i*lda+i+1] = -1
|
||
}
|
||
for j := i + 2; j < n; j++ {
|
||
a[i*lda+j] = 0
|
||
}
|
||
}
|
||
case blas.Lower:
|
||
for i := 0; i < n; i++ {
|
||
for j := 0; j < i-1; j++ {
|
||
a[i*lda+j] = 0
|
||
}
|
||
if i > 0 {
|
||
a[i*lda+i-1] = -1
|
||
}
|
||
a[i*lda+i] = tscal
|
||
}
|
||
}
|
||
dlarnv(b, 2, rnd)
|
||
case 16:
|
||
// One zero diagonal element.
|
||
diag = blas.NonUnit
|
||
switch uplo {
|
||
case blas.Upper:
|
||
for i := 0; i < n; i++ {
|
||
dlarnv(a[i*lda+i:i*lda+n], 2, rnd)
|
||
a[i*lda+i] = math.Copysign(2, a[i*lda+i])
|
||
}
|
||
case blas.Lower:
|
||
for i := 0; i < n; i++ {
|
||
dlarnv(a[i*lda:i*lda+i+1], 2, rnd)
|
||
a[i*lda+i] = math.Copysign(2, a[i*lda+i])
|
||
}
|
||
}
|
||
iy := n / 2
|
||
a[iy*lda+iy] = 0
|
||
dlarnv(b, 2, rnd)
|
||
bi.Dscal(n, 2, b, 1)
|
||
case 17:
|
||
// Make the offdiagonal elements large to cause overflow when
|
||
// adding a column of T. In the non-transposed case, the matrix
|
||
// is constructed to cause overflow when adding a column in
|
||
// every other step.
|
||
diag = blas.NonUnit
|
||
tscal := (1 - ulp) / (dlamchS / ulp)
|
||
texp := 1.0
|
||
switch uplo {
|
||
case blas.Upper:
|
||
for i := 0; i < n; i++ {
|
||
for j := i; j < n; j++ {
|
||
a[i*lda+j] = 0
|
||
}
|
||
}
|
||
for j := n - 1; j >= 1; j -= 2 {
|
||
a[j] = -tscal / float64(n+1)
|
||
a[j*lda+j] = 1
|
||
b[j] = texp * (1 - ulp)
|
||
a[j-1] = -tscal / float64(n+1) / float64(n+2)
|
||
a[(j-1)*lda+j-1] = 1
|
||
b[j-1] = texp * float64(n*n+n-1)
|
||
texp *= 2
|
||
}
|
||
b[0] = float64(n+1) / float64(n+2) * tscal
|
||
case blas.Lower:
|
||
for i := 0; i < n; i++ {
|
||
for j := 0; j <= i; j++ {
|
||
a[i*lda+j] = 0
|
||
}
|
||
}
|
||
for j := 0; j < n-1; j += 2 {
|
||
a[(n-1)*lda+j] = -tscal / float64(n+1)
|
||
a[j*lda+j] = 1
|
||
b[j] = texp * (1 - ulp)
|
||
a[(n-1)*lda+j+1] = -tscal / float64(n+1) / float64(n+2)
|
||
a[(j+1)*lda+j+1] = 1
|
||
b[j+1] = texp * float64(n*n+n-1)
|
||
texp *= 2
|
||
}
|
||
b[n-1] = float64(n+1) / float64(n+2) * tscal
|
||
}
|
||
case 18:
|
||
// Generate a unit triangular matrix with elements between -1
|
||
// and 1, and make the right hand side large so that it requires
|
||
// scaling. The diagonal is set to NaN.
|
||
diag = blas.Unit
|
||
switch uplo {
|
||
case blas.Upper:
|
||
for i := 0; i < n; i++ {
|
||
a[i*lda+i] = math.NaN()
|
||
dlarnv(a[i*lda+i+1:i*lda+n], 2, rnd)
|
||
}
|
||
case blas.Lower:
|
||
for i := 0; i < n; i++ {
|
||
dlarnv(a[i*lda:i*lda+i], 2, rnd)
|
||
a[i*lda+i] = math.NaN()
|
||
}
|
||
}
|
||
// Set the right hand side so that the largest value is bignum.
|
||
dlarnv(b, 2, rnd)
|
||
iy := bi.Idamax(n, b, 1)
|
||
bnorm := math.Abs(b[iy])
|
||
bscal := bignum / math.Max(1, bnorm)
|
||
bi.Dscal(n, bscal, b, 1)
|
||
case 19:
|
||
// Generate a triangular matrix with elements between
|
||
// bignum/(n-1) and bignum so that at least one of the column
|
||
// norms will exceed bignum.
|
||
// Dlatrs cannot handle this case for (typically) n>5.
|
||
diag = blas.NonUnit
|
||
tleft := bignum / math.Max(1, float64(n-1))
|
||
tscal := bignum * (float64(n-1) / math.Max(1, float64(n)))
|
||
switch uplo {
|
||
case blas.Upper:
|
||
for i := 0; i < n; i++ {
|
||
dlarnv(a[i*lda+i:i*lda+n], 2, rnd)
|
||
for j := i; j < n; j++ {
|
||
aij := a[i*lda+j]
|
||
a[i*lda+j] = math.Copysign(tleft, aij) + tscal*aij
|
||
}
|
||
}
|
||
case blas.Lower:
|
||
for i := 0; i < n; i++ {
|
||
dlarnv(a[i*lda:i*lda+i+1], 2, rnd)
|
||
for j := 0; j <= i; j++ {
|
||
aij := a[i*lda+j]
|
||
a[i*lda+j] = math.Copysign(tleft, aij) + tscal*aij
|
||
}
|
||
}
|
||
}
|
||
dlarnv(b, 2, rnd)
|
||
bi.Dscal(n, 2, b, 1)
|
||
}
|
||
|
||
// Flip the matrix if the transpose will be used.
|
||
if trans == blas.Trans {
|
||
switch uplo {
|
||
case blas.Upper:
|
||
for j := 0; j < n/2; j++ {
|
||
bi.Dswap(n-2*j-1, a[j*lda+j:], 1, a[(j+1)*lda+n-j-1:], -lda)
|
||
}
|
||
case blas.Lower:
|
||
for j := 0; j < n/2; j++ {
|
||
bi.Dswap(n-2*j-1, a[j*lda+j:], lda, a[(n-j-1)*lda+j+1:], -1)
|
||
}
|
||
}
|
||
}
|
||
|
||
return diag
|
||
}
|
||
|
||
func checkMatrix(m, n int, a []float64, lda int) {
|
||
if m < 0 {
|
||
panic("testlapack: m < 0")
|
||
}
|
||
if n < 0 {
|
||
panic("testlapack: n < 0")
|
||
}
|
||
if lda < max(1, n) {
|
||
panic("testlapack: lda < max(1, n)")
|
||
}
|
||
if len(a) < (m-1)*lda+n {
|
||
panic("testlapack: insufficient matrix slice length")
|
||
}
|
||
}
|