From 96745f0db0280a2d5c6b591950c22a30e7e8c26b Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Wed, 24 Jul 2024 11:17:26 -0400 Subject: [PATCH 01/23] Fix self-induced git problem --- Joss_Paper_Backup/draft-pdf.yml | 23 ++++++++++++++++++++ {Joss_Paper => Joss_Paper_Backup}/paper.bib | 0 {Joss_Paper => Joss_Paper_Backup}/paper.md | 0 {Joss_Paper => Joss_Paper_Backup}/paper.pdf | Bin 4 files changed, 23 insertions(+) create mode 100644 Joss_Paper_Backup/draft-pdf.yml rename {Joss_Paper => Joss_Paper_Backup}/paper.bib (100%) rename {Joss_Paper => Joss_Paper_Backup}/paper.md (100%) rename {Joss_Paper => Joss_Paper_Backup}/paper.pdf (100%) diff --git a/Joss_Paper_Backup/draft-pdf.yml b/Joss_Paper_Backup/draft-pdf.yml new file mode 100644 index 00000000..8573b798 --- /dev/null +++ b/Joss_Paper_Backup/draft-pdf.yml @@ -0,0 +1,23 @@ +on: [push] + +jobs: + paper: + runs-on: ubuntu-latest + name: Paper Draft + steps: + - name: Checkout + uses: actions/checkout@v4 + - name: Build draft PDF + uses: openjournals/openjournals-draft-action@master + with: + journal: joss + # This should be the path to the paper within your repo. + paper-path: Joss_Paper/paper.md + - name: Upload + uses: actions/upload-artifact@v1 + with: + name: paper + # This is the output path where Pandoc will write the compiled + # PDF. Note, this should be the same directory as the input + # paper.md + path: Joss_Paper diff --git a/Joss_Paper/paper.bib b/Joss_Paper_Backup/paper.bib similarity index 100% rename from Joss_Paper/paper.bib rename to Joss_Paper_Backup/paper.bib diff --git a/Joss_Paper/paper.md b/Joss_Paper_Backup/paper.md similarity index 100% rename from Joss_Paper/paper.md rename to Joss_Paper_Backup/paper.md diff --git a/Joss_Paper/paper.pdf b/Joss_Paper_Backup/paper.pdf similarity index 100% rename from Joss_Paper/paper.pdf rename to Joss_Paper_Backup/paper.pdf From e885bef9fb2bb7bf00dbe48a8f95d09f63aeb792 Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Wed, 24 Jul 2024 11:38:38 -0400 Subject: [PATCH 02/23] skip ci --- {Joss_Paper_Backup => Joss_Paper}/paper.bib | 0 {Joss_Paper_Backup => Joss_Paper}/paper.md | 0 {Joss_Paper_Backup => Joss_Paper}/paper.pdf | Bin Joss_Paper_Backup/draft-pdf.yml | 23 -------------------- 4 files changed, 23 deletions(-) rename {Joss_Paper_Backup => Joss_Paper}/paper.bib (100%) rename {Joss_Paper_Backup => Joss_Paper}/paper.md (100%) rename {Joss_Paper_Backup => Joss_Paper}/paper.pdf (100%) delete mode 100644 Joss_Paper_Backup/draft-pdf.yml diff --git a/Joss_Paper_Backup/paper.bib b/Joss_Paper/paper.bib similarity index 100% rename from Joss_Paper_Backup/paper.bib rename to Joss_Paper/paper.bib diff --git a/Joss_Paper_Backup/paper.md b/Joss_Paper/paper.md similarity index 100% rename from Joss_Paper_Backup/paper.md rename to Joss_Paper/paper.md diff --git a/Joss_Paper_Backup/paper.pdf b/Joss_Paper/paper.pdf similarity index 100% rename from Joss_Paper_Backup/paper.pdf rename to Joss_Paper/paper.pdf diff --git a/Joss_Paper_Backup/draft-pdf.yml b/Joss_Paper_Backup/draft-pdf.yml deleted file mode 100644 index 8573b798..00000000 --- a/Joss_Paper_Backup/draft-pdf.yml +++ /dev/null @@ -1,23 +0,0 @@ -on: [push] - -jobs: - paper: - runs-on: ubuntu-latest - name: Paper Draft - steps: - - name: Checkout - uses: actions/checkout@v4 - - name: Build draft PDF - uses: openjournals/openjournals-draft-action@master - with: - journal: joss - # This should be the path to the paper within your repo. - paper-path: Joss_Paper/paper.md - - name: Upload - uses: actions/upload-artifact@v1 - with: - name: paper - # This is the output path where Pandoc will write the compiled - # PDF. Note, this should be the same directory as the input - # paper.md - path: Joss_Paper From 5d1f27421c919410c7790182077833fcb2875d27 Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Wed, 24 Jul 2024 11:42:41 -0400 Subject: [PATCH 03/23] Fix stuff --- .github/workflows/draft-pdf.yml | 23 +++++++++++++++++++++++ 1 file changed, 23 insertions(+) create mode 100644 .github/workflows/draft-pdf.yml diff --git a/.github/workflows/draft-pdf.yml b/.github/workflows/draft-pdf.yml new file mode 100644 index 00000000..8573b798 --- /dev/null +++ b/.github/workflows/draft-pdf.yml @@ -0,0 +1,23 @@ +on: [push] + +jobs: + paper: + runs-on: ubuntu-latest + name: Paper Draft + steps: + - name: Checkout + uses: actions/checkout@v4 + - name: Build draft PDF + uses: openjournals/openjournals-draft-action@master + with: + journal: joss + # This should be the path to the paper within your repo. + paper-path: Joss_Paper/paper.md + - name: Upload + uses: actions/upload-artifact@v1 + with: + name: paper + # This is the output path where Pandoc will write the compiled + # PDF. Note, this should be the same directory as the input + # paper.md + path: Joss_Paper From a0344b0768d2939af4d9ec909bb29f95b9a871bb Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Wed, 24 Jul 2024 11:58:17 -0400 Subject: [PATCH 04/23] Update paper with solve data --- Joss_Paper/paper.md | 12 ++++++++++-- 1 file changed, 10 insertions(+), 2 deletions(-) diff --git a/Joss_Paper/paper.md b/Joss_Paper/paper.md index 456b3070..8e669a0d 100644 --- a/Joss_Paper/paper.md +++ b/Joss_Paper/paper.md @@ -104,7 +104,7 @@ GMRES and Bi-CGSTAB solvers for Krylov-IR methods are taken from Here is a simple example to show how ```mplu``` works. We will follow that with some benchmarking on the cost of factorizations. -The computations were done with Julia 1.10.2 +The computations were done with Julia 1.10.4 on an Apple Mac Mini with an M2 pro processor and 32GB RAM. We used OPENBLAS for LAPACK and the BLAS for this example. Other choices, such as the [AppleAccelerate](https://github.com/JuliaLinearAlgebra/AppleAccelerate.jl) @@ -198,7 +198,10 @@ neither ```mplu``` nor ```lu!``` allocate storage for a new high precision array, but mplu allocates for a low precision copy, so the memory and allocation cost for mplu is -50\% more than lu. +50\% more than lu. One issue with smaller problems is that the triangular solve +does not parallelize as well as the factorization, so does not exploit +multi-core processor as well. We can see this in the IR solver times because +each iteration of IR needs a matrix-vector multiply and a triangular solve. ``` julia> using BenchmarkTools @@ -208,6 +211,11 @@ julia> @belapsed mplu($A) julia> @belapsed lu!(AC) setup=(AC=copy($A)) 1.42840e-01 + +# And now for the solve times + + + ``` # A Few Subtleties From 6fe26178037f270529ac4fad726560cea6febbb7 Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Wed, 24 Jul 2024 12:01:09 -0400 Subject: [PATCH 05/23] skip ci --- Joss_Paper/paper.pdf | Bin 239221 -> 239510 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/Joss_Paper/paper.pdf b/Joss_Paper/paper.pdf index ea13f206fed5e2a002980630bd33f8941680c6df..ece33e9e9e457eb4eb79fb196760f7f82cbd397c 100644 GIT binary patch delta 38962 zcmV)EK)}EC%np{%4uG@)0#%p5S_2ig7gYhU2?3^;$X)?Ae}yXL1zpCe%+n;V`#Or8 z7}8^Ie7zup3eHsW+{&V*_WmB_Ewwf}C4td+vvp2%NR@~lAo&0)`T;m2d4#I<*Lny` zKau^2*n>R8&viWbZ*fdAu#e>7m6W5hfyp3R8{E?%tk$T6weoY#cP&}c3C<+in=WK- zbwjjaNiZ8}e~I-;rFvy{rY>cvjXM4;Y}u;+AKrFjPYPoXD+%LrHVjcf0Hthu)dLTJYBlkA9@pA~ z{PwhGJFJe;@N;F7BQ`29U5^IJy%4^7af2ULbHpB@f2rLHxfP@ORwUNlloZ`_`^=dl zw{4LMKe#8b?`uNHD!d#CBfg@eiD_9(!(#Ftl(LjxhmNP_j!8F8NIF7HMY-HJC#>0H zuJk*XH_9~4HwD5&G;I#yPRi;)Lia){lVnvornT?xhpgXK*yXSYhC%o}PPfKkFxXPw z4mm6#e}tXD+bDA6Zm)Eachy#Wx&p*qYgF!~CA-7iua2n6(wxdKr(m@tH*TU%7)!wu zZ)^6aLza9vP;i^esz()3A)aycL-`gR$&_m4Yh!cI*7A<5cdt}oDuEZpRD0E=*t1$R zVVZ5Gn+qxC;c7^?U6O(BX4%lT4M_6nxl&JVe^q=8p5$&Vk1~cy=?5C9o8eb(cFV++ z^~?neBf&O{lp+-M#BysvP|w(*pW6;wBLA9d*@<8FX&`7--x(XJF?M2p`Wvs6raS7`7e zo=ay1`!IW31cQVPw}N9T`wl({+9BSs(7f~R=Rsnk-<$?s!lSYg%e>aj4M#6-r2_r3MZeHYcB4z~EJ$Wt?^vE%#fOvLY z>6`Y~H$waC8{KW|s?p6xyjG$Uy>TsdQ}a90{2h>0JM7>my=U5r5x=UOO?QdRuBmD~ zp#@`WeB?ztJvbW-_R%;!9*R&J6md(3B-;nKWQYziQ66BBa>8V_3Wx^k6*-ltxR5&PQ<^M0WrYFeLLrdh@ zOS?*@0&a;uYkxfRbzkwkJXTD^+Fm-($CFh4+!?LGOR(Sx_v_NRu}L2hr57Z& z%Y%95y}ls7y=$_Z^eJ^DOLBC0qPtIhn#vG7sayM$dGevH+xcybqP>)0!iE8?qqK(6 z4#J#Km24*5LA(72Fpr({mtAE6JOeW_G?O6<6@TSiOONEZ4ZioUs6WseiPRHlG*DfS zAbXpnZ;L%`kX$kdl1u)6s0U?9c2&95b66l46xx<0iXuNgEK?&7lK%bYAL)-g{N+zC za@C^Jg*qs!1S)+1rGyD^`2EMrzh8utPKHDI^V;9Wanwr+f8bvv+}t+gUzz@EGr`B_ z%YV14LYiEA;gt`E&%fVf=<~)fqssz&HHbkzice*@_BD<-#13w7S5~E zD8TxIF_Mz2b#uv;V}&v}l91OMK>2nJ;h;ru zGInl+VHD|lMMru1!A)8tQ$oV1DW!$;3!tw_Q{bbXJZVg;dH8CkRFT7rItu0rvJnwJx)9e<+K=ZrsiVE%OxK(zwbWx7acs%6)8i)nFz- z(M!0&*+!T~Vl1UTJyfwHDViStd23ny(0b+_lBo@AMMFhIWsk~+^F!&e@Q5Gd->hUi2{3yyw zRPF;CNe)Kg$Ugc^MqO_LyGj5-3awWXsIcuEPr|28FP92>}R!93pAps}$iNBaAUgGt6R(SndlJMMa;jO6YKta-X)chU%+!r&2Y3cBfCFb;E zM);z8pW9mx73s~ZqF-+kUYJ(}7@<_JisQ5@6bfa(tO{8~6kC(nPs{606!Wp9qe%rm z)1Mb-&$A->O3jN?Du42}zM}@S;xs-8FgwMGA;k!wBn03?0=+!|Qj9ROIr}hsS_)1J z#yBS+xQ?2h7K}9%y5u)n9RJ;4GQ$V(CxI)wGtZl2|eymq>QECsYsbiLTc0o4(69&|?$c)}BB(K-vFop~Nv4oZwmQwqIxfGBpiW$!kWP|KUA@?guVwn#!l&}h8 zZAWc7%;7NT<#0%3y2&zQpYa^5$)-(u^qMPAfjy$#P-v^^1P%zjJ*Y& z^x{#ja(j;CXMg0@Q;F0OtzIoj&xfiW6dCb6Pt6N(w1r5U<9y_u)3e@J_qI?ybffl? zvN6AnGn!R8@4OF&dGx&D zbZn+2&xbpWC_I{u+H|T|Jxlc@xm%xy@ zyIdBdZQfOid8lFvfmIrPKJAAL5>XHX|6K5AiWY>DdPQ+M0YD>#(tAU}B?G!OiTtsn4FRMM4gq72 zNip_3)RC$NRyF9nE zb-$#sm;S9{V>5k1rMUX`Q)ZP%)t&P%{(VdPFje#(GxuL1uQ!^N){ z?tf3P+Ch#`d$%tK}~_{IOx#s@T|WXjIsK5t>dC2cD}C z=ChkvR%V-RVx~Bq-GH>h`kn7fyt}gqGxwzHh)amYAj)pJH&RL6dONxMK~;NJFJ zFm>}T7W>?Hjy$uvG4qz)YTxpDuP3c|Vt*HjmD$U_D$~p=lk#5Sn*H&+s)z!K-;9hb z-;VhE5vFfQajDTs0i5#pq(vy{>ypfWV{f3m12dIwRHn}0*{>>?;!Hio*`&%w3A4)V zY!QTVR=7~JWz=20>bj2RUfC7b7cEspb==*utK8!!mR+&1`n6SIx!KYy1NNXytbYTl zo2A%^NNvt|=UeCSa*kiGeS}&*86>3kEgMS(%7#q92R`Ncl9kp0S3A8H`ayGFT=v2${&sbI% zElXPwmTy^`HL4kSrKrB3u*fyv)k?@HY56vIo9b3{Q`%_T!HJyb!KNvVtm=gV@@1TB zizm6Ycb;;(;IcZvJH`M)2R#aKuQy;lXfy)6AIaR~x)D6oRi1}J`Ya9AR)6Cb4HQG^ zic7P88gt4rFZ7P@UVjO={n{#U!xO+&s~Czjqv_mdj-hfl4BVc&+brnfSu!*$&FWoh z>AoGGH@>Lz#_ab}`NU-@VsH6Mo;=Ur&$OsLa~m$Fri{NWgTHA|4Q*_Uy4vV+Xojca zO8a9g-p5hXC~~CAjr!VYCx2@3$ASD6jAZn+^8%Cj>}ghfPVZ1CKr(UlM9`=Sk9PvK z8Y!&&soV0WwT}O;_jI_Yp^F!(R&P7gwuh(?mD!8)?vqDcS)gAFiiGP}0Y746OqIHBK zZp-3E)q2lcL3Hp?e(gn;Gj455tC}#XraRssx7_TVNWa2E?~+rW_%0mx8fCYTE(5Tu zMMLNK&l!iwS(``!C0j>7XY+bipRCSZ<(YI5Xd|G;lTJXWS9gWbGY9pg9WWGvkYULS zRiku1gmP{`q1T~GZGXOOY=`IMu}UbeQ1Z>25HvgqjW zP)hpC`GIY*y-;0_@iiwDhkokkCa32vw#(Jh{j&5qO!exokb34ApVR}SD7LAkU}xj4 zp&Xd3W-Zx#Y7TYjcR=D(d11#f?t#d`clVLgHdZS~bPbC0?|)UI42~sz!}b(s^<&kK zR5jJpb+~zEw7#%5qgMMI@!Tn5@3vbT;MZlj+VSXS!3HARDIXtTWpwH=38Y@>6-avb zZusRho+;-8VL_lY2kQh7-hTh_^6PKZ{x|yla*zUaIQ%#D{6Tpq3T@)Je;)q&@;#p; 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z4aH<;k1+}>K<&?MVm%)2_v)I_Jtiv@%Ts&g>u^CnqwPA18X`dRdv|@L^Re|}8fHS! zm!hX~@}YPE(s+I7D}VcoiRzs*TvjKyQbK1h+{`pf?~rxEcvaY*=TDguZ6f|;DDM3! zYG{v^4SV>QwLPo(#f^7ch8E9JAv(@mqj7b%Y;o(nbtMLt08iKK>;m;ngPSG2!eh4n zrhV@j$q(2LWieG~K>sFqy$4Dj1N|z2lhGs-f!;=^d{WfD_so5=m{Y2Cqk&yn2e(g* zAY>Wx^G%H?5qc}})BKTGX6N45`KhMJ<*{saJ-hhYgXU;vhYjfT0$)kDUt$HZ8!>#1 zh&t0?3QkWkepx}AawHi40-B?n3i2^u@Fc`L{Uy ziDtjJ_iT665nsJx;$?bGk1-Dy=ZQ{~yh5@szYr(In7klZW*lbjNZpEVCpxTLb8+D{ zA2fPzDV??!lOp=&s%5z_&yN}1mnVE48>e5DRN-nCNj>Z{cW<)tFm&2W%g|RdOu05V gStGq^N54(Oi_a_uJ@Wt&W{fh5SzKKAlpgc{01RV-2mk;8 From eff00474efc435c5aff086bf6804f7474c3a0d83 Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Wed, 24 Jul 2024 12:02:53 -0400 Subject: [PATCH 06/23] update to v4 --- .github/workflows/draft-pdf.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.github/workflows/draft-pdf.yml b/.github/workflows/draft-pdf.yml index 8573b798..902c710a 100644 --- a/.github/workflows/draft-pdf.yml +++ b/.github/workflows/draft-pdf.yml @@ -14,7 +14,7 @@ jobs: # This should be the path to the paper within your repo. paper-path: Joss_Paper/paper.md - name: Upload - uses: actions/upload-artifact@v1 + uses: actions/upload-artifact@v4 with: name: paper # This is the output path where Pandoc will write the compiled From 370abfc893fd06bd9d1cf14e695437470f2aa330 Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Thu, 25 Jul 2024 14:35:29 -0400 Subject: [PATCH 07/23] Update revised paper --- Joss_Paper/paper.md | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/Joss_Paper/paper.md b/Joss_Paper/paper.md index 8e669a0d..4a04fe26 100644 --- a/Joss_Paper/paper.md +++ b/Joss_Paper/paper.md @@ -214,10 +214,16 @@ julia> @belapsed lu!(AC) setup=(AC=copy($A)) # And now for the solve times +julia> @belapsed ldiv!($AF,bb) setup=(bb = copy($b)) +4.79117e-03 - +julia> @belapsed $MPF\$b +2.01195e-02 ``` +So the total solve time (factorization and solve) is less for IR +by roughly 40%. + # A Few Subtleties Within the 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z=;=vdDvLenq&pW56cvOxEf-EC~9=z!^d*q{Ag@aaBpM(*CMyLywUn&6+@&|DdW0)u>(dE=O6{IuuE~>9rKx z+roG4^rvA7JTiO81hyGs1I)#c_wl>Q6!Q*SqQ-ezEMgB~T{Q|>Z6wsW<{(Qe^asRN zF%1lB_1Gdj$T9mQX@V4L@jU~yy2twKUykE z#|ETpaG6Z!m2V2e0TZK=##K@vQ{GQC`IzV>4^xgXtri{7RUtk)aYfsz39Fk(;_Nlm zbz9hJR^%V7L|(EBP~<7*NwxS@+wknw`O%}=2fB~_$QL<0mBa~(%Ck~RY}7C6!rN?X zX;h8)Kd=Bd?04)k1aY}K9t_5Ij*=}xN+8OwhP?jR=v~h!$zxvH*N3Y;lqkdy9 xYu7Kw2ydRV5188I46gr2;rU+@j+?uQtGl Date: Tue, 24 Sep 2024 15:06:14 -0400 Subject: [PATCH 09/23] Paper update --- Joss_Paper/old_paper.md | 303 ++++++++++++++++++++++++++++++++++++++++ Joss_Paper/paper.md | 124 ---------------- 2 files changed, 303 insertions(+), 124 deletions(-) create mode 100644 Joss_Paper/old_paper.md diff --git a/Joss_Paper/old_paper.md b/Joss_Paper/old_paper.md new file mode 100644 index 00000000..4a04fe26 --- /dev/null +++ b/Joss_Paper/old_paper.md @@ -0,0 +1,303 @@ +--- +title: 'MultiPrecisionArrays.jl: A Julia package for iterative refinement' +tags: + - Julia + - mathematics + - numerical analysis + - linear equations +authors: + - name: C. T. Kelley + orcid: 0000-0003-2791-0648 + affiliation: 1 +affiliations: + - name: North Carolina State University, Raleigh NC, USA + index: 1 +date: 15 April 2024 +bibliography: paper.bib +--- + +# Summary + +[MultiPrecisionArrays.jl](https://github.com/ctkelley/MultiPrecisionArrays.jl), +[@kelley:2024a; @kelley:2024b] provides data structures and +solvers for several variations of iterative refinement (IR). IR can speed up an LU matrix factorization for solving linear systems of equations by +factoring a low precision copy of the matrix and using that +low precision factorization in a iteration loop to solve the system. +For example, if high precision is double and low precision is single, +then the factorization time is cut in half. +The additional storage cost is the low precision copy, +so IR is at time vs storage trade off. IR has a long history +and a good account of the classical theory is in [@higham:1996]. + +# Statement of need + +The solution of linear systems of equations is a ubiquitous task +in computational science and engineering. A common method for dense +systems is Gaussian elimination done via an LU factorization, +[@higham:1996]. Iterative refinement is a way to reduce the factorization +time at the cost of additional storage. +[MultiPrecisionArrays.jl](https://github.com/ctkelley/MultiPrecisionArrays.jl) +enables IR with a simple interface in Julia [@Juliasirev:17] with an +IR factorization object that one uses in the same way as the one for LU. +The package offers several variants of IR, both classical +[@Wilkinson:48; @higham:1996], and some from the recent literature +[@CarsonHigham:2017; @amestoy:2024]. + +# Algorithm + +This package will make solving dense systems of linear equations faster by using the LU factorization and IR. While other factorizations can be used in IR, the +package is limited to LU for now. A very generic description of this +for solving a linear system $A x = b$ in a high (working) precision is + +__IR(A, b)__ + +- $x = 0$ + +- $r = b$ + +- Factor $A = LU$ in a lower precision + +- While $\| r \|$ is too large + + - $d = (LU)^{-1} r$ + + - $x = x + d$ + + - $r = b - Ax$ + +- end + +- end + + +In Julia, a code to do this would solve the linear system $A x = b$ in +in the working precision, say double, +by using a factorization in a lower (factorization) +precision, say single, within a residual correction iteration. +This means that one would need to allocate storage for a copy of $A$ +in the factorization precision and factor that copy. + +The multiprecision factorization ```mplu``` makes the low precision copy +of the matrix, factors that copy, and allocates some storage for the +iteration. The original matrix and the low precision factorization +are stored in a factorization object that you can use with ```\```. + +IR is a perfect example of a storage/time trade off. +To solve a linear system $A x = b$ in $R^N$ with IR, +one incurs the storage penalty of making a low +precision copy of $A$ and reaps the benefit of only having to +factor the low precision copy. + +# Installation + +The standard way to install a package is to type ```import.Pkg; Pkg.add("MultiPrecisionArrays")``` at the Julia prompt. One can run the unit tests with ```Pkg.test("MultiPrecisionArrays")```. +After installation, type ```using MultiPrecisionArrays``` when you want to use the functions in the package. + +There are only two direct dependencies outside of the Julia standard libraries. +The factorization in half precision (Float16) uses +[OhMyThreads.jl](https://github.com/JuliaFolds2/OhMyThreads.jl). The +GMRES and Bi-CGSTAB solvers for Krylov-IR methods are taken from +[SIAMFANL.jl](https://github.com/ctkelley/SIAMFANLEquations.jl) +[@kelley:2022b]. + +# Example + +Here is a simple example to show how ```mplu``` works. +We will follow that with some benchmarking on the cost of factorizations. +The computations were done with Julia 1.10.4 +on an Apple Mac Mini with an M2 pro processor and 32GB RAM. We used +OPENBLAS for LAPACK and the BLAS for this example. Other choices, such as the +[AppleAccelerate](https://github.com/JuliaLinearAlgebra/AppleAccelerate.jl) +framework would work equally well. + +In this example high (working) +precision is double, ```Float64```, and low (factorization) +precision is single, ```Float32```. +The matrix is the sum of the identity and a constant multiple of the trapezoid rule discretization of the Greens operator for $-d^2/dx^2$ on $[0,1]$ + +$$ +G u(x) = \int_0^1 g(x,y) u(y) \, dy +$$ + +where + +$$ +g(x,y) = \min(x,y) ( 1 - \max(x,y) ). +$$ + +The discretization for an $N$-point discretization is the $N \times N$ +maatrix +$$ +G_{ij} = g(x_i, x_j) /(N+1) +$$ +where $x_i = i/(N+1)$. + + +The code for construction $G$ is in the __/src/Examples__ directory. The file is __Gmat.jl__. You need to do +``` +using MultiPrecisionArrays +using MultiPrecisionArrays.Examples +``` +to use ```mplu``` and ```Gmat```. + +Now we will see how the results look. In this example we compare the result with iterative refinement with ```A\b```, which is LAPACK's LU. +As you can see the results are equally good. +Note that a factorization object ```MPF``` is the +output of ```mplu```. This is analogous to ```AF=lu(A)``` in LAPACK. + +``` +julia> using MultiPrecisionArrays + +julia> using MultiPrecisionArrays.Examples + +julia> using BenchmarkTools + +julia> N=4096; + +julia> # Build G on an N x N grid + +julia> G=Gmat(N); + +julia> # The operator is I - G + +julia> A = I - G; + +julia> # Create a test problem where the solution is x = ones(N) + +julia> x=ones(N); b=A*x; + +julia> # Compute the mplu and lu factorizations of A + +julia> MPF=mplu(A); AF=lu(A); + +julia> # Solve A x = b with both factorizations + +julia> z=MPF\b; w=AF\b; + +julia> # Compute relative errors and residuals for mplu + +julia> mpluerr=norm(z-x,Inf); mpluresid=norm(b-A*z,Inf)/norm(b,Inf); + +julia> # Compute relative errors and residuals for lu + +julia> luerr=norm(w-x,Inf); luresid=norm(b-A*w,Inf)/norm(b,Inf); + +julia> # Print the results + +julia> println("Errors: $mpluerr, $luerr. Residuals: $mpluresid, $luresid") +Errors: 4.44089e-16, 6.68354e-14. Residuals: 4.44089e-16, 6.68354e-14 +``` + +So the results are equally good. + +The compute time for ```mplu``` should be roughly half that of ```lu```. +A fair comparison is with ```lu!```, which does not allocate new storage +for the factorization. ```mplu``` factors a low precision array, +so the factorization cost is cut in half. Memory is a different story because +neither ```mplu``` nor ```lu!``` +allocate storage for a new high precision array, +but mplu allocates for a low precision copy, +so the memory and allocation cost for mplu is +50\% more than lu. One issue with smaller problems is that the triangular solve +does not parallelize as well as the factorization, so does not exploit +multi-core processor as well. We can see this in the IR solver times because +each iteration of IR needs a matrix-vector multiply and a triangular solve. + +``` +julia> using BenchmarkTools + +julia> @belapsed mplu($A) +8.60945e-02 + +julia> @belapsed lu!(AC) setup=(AC=copy($A)) +1.42840e-01 + +# And now for the solve times + +julia> @belapsed ldiv!($AF,bb) setup=(bb = copy($b)) +4.79117e-03 + +julia> @belapsed $MPF\$b +2.01195e-02 +``` + +So the total solve time (factorization and solve) is less for IR +by roughly 40%. + +# A Few Subtleties + +Within the algorithm one has to determine what the line +$d = (LU)^{-1} r$ means. Does one cast $r$ into the lower precision before +the solve or not? If one casts $r$ into the lower precision, then the solve +is done entirely in the factorization precision. If, however, $r$ remains +in the working precision, then the LU factors are promoted to the working +precision on the fly. This makes little difference if TW is double and +TF is single and there is a modest performance benefit to downcasting $r$ into +single. Therefore that is the default behavior in that case. +If TF is half precision, ```Float16```, then it is best to do the interprecision +transfers on the fly and if +one is using one of the Krylov-IR algorithms +[@amestoy:2024] then one must do the interprecision transfers on the fly +and not downcast $r$. + +There are two half precision (16 bit) formats. Julia has native support for IEEE 16 bit floats (Float16). A second format (BFloat16) has a larger exponent field and a smaller significand (mantissa), thereby trading precision for range. In fact, the exponent field in BFloat is the same size (8 bits) as that for single precision (Float32). The significand, however, is only 8 bits. Compare this to the size of the exponent fields for Float16 (11 bits) and single (24 bits). The size of the significand means that you can get in real trouble with half precision in either format and that IR is more likely to fail to converge. GMRES-IR +can mitigate the convergence problems [@amestoy:2024] by using the +low-precision solve as a preconditioner. We support both GMRES +[@saad:1986] and BiCGSTAB [@VanderVorst:1992] as solvers for +Krylov-IR methods. One should also know that LAPACK and the BLAS do not yet +support half precision arrays, so working in Float16 will be slower than +using Float64. + +The classic algorithm from [@Wilkinson:48] and its recent extension +[@CarsonHigham:2017] evaluate the residual in a higher precision +that the working precision. This can give improved accuracy for +ill-conditioned problems at a cost of the interprecision transfers +in the residual computation. This needs to be implemented with some +care and [@demmel:06] has an excellent account of the details. + +__MultiPrecisionArrays.jl__ provides +infrastructure to manage these things and we refer the reader to +[@kelley:2024b] for the details. + +# Projects using __MultiPrecisionArrays.jl__. + +This package was motivated by +the use of low-precision factorizations in Newton's method +[@kelley:2022a, @kelley:2022b] and the interface between +a preliminary version of this +package and the solvers from [@kelley:2022c] was reported in +[@kelley:2023]. That paper used a three +precision form of IR (TF=half, TW=single, nonlinear residual +computed in double) and required direct use of multiprecision +constructors that we do not export in __MultiPrecisionArrays.jl__. +We will fully support the application to nonlinear solvers in +a future version. We give a detailed account of interprecision +transfers in [@kelley:2024c] and use +__MultiPrecisionArrays.jl__ to generate the table in that paper. + + +# Other Julia Packages for IR + +The package +[IterativeRefinement.jl](https://github.com/RalphAS/IterativeRefinement.jl) +is an implementation of the IR method from +[@dongarra:1983]. It has not been updated in four years. + +The unregistered +package [Itref.jl](https://github.com/bvieuble/Itref.jl) implements +IR and the GMRES-IR method from [@amestoy:2024] and was used to obtain +the numerical results in that paper. It does not provide the +data structures for preallocation that we do and does not seem to have +been updated lately. + + +# Acknowledgements + +This work was partially supported by +Department of Energy grant DE-NA003967. +Any opinions, findings, and conclusions or recommendations +expressed in this material are those of the author and +do not necessarily reflect the views of the United States Department of Energy. + + +# References diff --git a/Joss_Paper/paper.md b/Joss_Paper/paper.md index 4a04fe26..3056b8ad 100644 --- a/Joss_Paper/paper.md +++ b/Joss_Paper/paper.md @@ -100,130 +100,6 @@ GMRES and Bi-CGSTAB solvers for Krylov-IR methods are taken from [SIAMFANL.jl](https://github.com/ctkelley/SIAMFANLEquations.jl) [@kelley:2022b]. -# Example - -Here is a simple example to show how ```mplu``` works. -We will follow that with some benchmarking on the cost of factorizations. -The computations were done with Julia 1.10.4 -on an Apple Mac Mini with an M2 pro processor and 32GB RAM. We used -OPENBLAS for LAPACK and the BLAS for this example. Other choices, such as the -[AppleAccelerate](https://github.com/JuliaLinearAlgebra/AppleAccelerate.jl) -framework would work equally well. - -In this example high (working) -precision is double, ```Float64```, and low (factorization) -precision is single, ```Float32```. -The matrix is the sum of the identity and a constant multiple of the trapezoid rule discretization of the Greens operator for $-d^2/dx^2$ on $[0,1]$ - -$$ -G u(x) = \int_0^1 g(x,y) u(y) \, dy -$$ - -where - -$$ -g(x,y) = \min(x,y) ( 1 - \max(x,y) ). -$$ - -The discretization for an $N$-point discretization is the $N \times N$ -maatrix -$$ -G_{ij} = g(x_i, x_j) /(N+1) -$$ -where $x_i = i/(N+1)$. - - -The code for construction $G$ is in the __/src/Examples__ directory. The file is __Gmat.jl__. You need to do -``` -using MultiPrecisionArrays -using MultiPrecisionArrays.Examples -``` -to use ```mplu``` and ```Gmat```. - -Now we will see how the results look. In this example we compare the result with iterative refinement with ```A\b```, which is LAPACK's LU. -As you can see the results are equally good. -Note that a factorization object ```MPF``` is the -output of ```mplu```. This is analogous to ```AF=lu(A)``` in LAPACK. - -``` -julia> using MultiPrecisionArrays - -julia> using MultiPrecisionArrays.Examples - -julia> using BenchmarkTools - -julia> N=4096; - -julia> # Build G on an N x N grid - -julia> G=Gmat(N); - -julia> # The operator is I - G - -julia> A = I - G; - -julia> # Create a test problem where the solution is x = ones(N) - -julia> x=ones(N); b=A*x; - -julia> # Compute the mplu and lu factorizations of A - -julia> MPF=mplu(A); AF=lu(A); - -julia> # Solve A x = b with both factorizations - -julia> z=MPF\b; w=AF\b; - -julia> # Compute relative errors and residuals for mplu - -julia> mpluerr=norm(z-x,Inf); mpluresid=norm(b-A*z,Inf)/norm(b,Inf); - -julia> # Compute relative errors and residuals for lu - -julia> luerr=norm(w-x,Inf); luresid=norm(b-A*w,Inf)/norm(b,Inf); - -julia> # Print the results - -julia> println("Errors: $mpluerr, $luerr. Residuals: $mpluresid, $luresid") -Errors: 4.44089e-16, 6.68354e-14. Residuals: 4.44089e-16, 6.68354e-14 -``` - -So the results are equally good. - -The compute time for ```mplu``` should be roughly half that of ```lu```. -A fair comparison is with ```lu!```, which does not allocate new storage -for the factorization. ```mplu``` factors a low precision array, -so the factorization cost is cut in half. Memory is a different story because -neither ```mplu``` nor ```lu!``` -allocate storage for a new high precision array, -but mplu allocates for a low precision copy, -so the memory and allocation cost for mplu is -50\% more than lu. One issue with smaller problems is that the triangular solve -does not parallelize as well as the factorization, so does not exploit -multi-core processor as well. We can see this in the IR solver times because -each iteration of IR needs a matrix-vector multiply and a triangular solve. - -``` -julia> using BenchmarkTools - -julia> @belapsed mplu($A) -8.60945e-02 - -julia> @belapsed lu!(AC) setup=(AC=copy($A)) -1.42840e-01 - -# And now for the solve times - -julia> @belapsed ldiv!($AF,bb) setup=(bb = copy($b)) -4.79117e-03 - -julia> @belapsed $MPF\$b -2.01195e-02 -``` - -So the total solve time (factorization and solve) is less for IR -by roughly 40%. - # A Few Subtleties Within the algorithm one has to determine what the line From 0ef4538b8f6a2fea9cfac5253b20e4e317c89e7d Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Tue, 24 Sep 2024 15:17:42 -0400 Subject: [PATCH 10/23] Update paper; remove example --- Joss_Paper/old_paper.md | 303 ---------------------------------------- Joss_Paper/paper.pdf | Bin 241198 -> 222085 bytes 2 files changed, 303 deletions(-) delete mode 100644 Joss_Paper/old_paper.md diff --git a/Joss_Paper/old_paper.md b/Joss_Paper/old_paper.md deleted file mode 100644 index 4a04fe26..00000000 --- a/Joss_Paper/old_paper.md +++ /dev/null @@ -1,303 +0,0 @@ ---- -title: 'MultiPrecisionArrays.jl: A Julia package for iterative refinement' -tags: - - Julia - - mathematics - - numerical analysis - - linear equations -authors: - - name: C. T. Kelley - orcid: 0000-0003-2791-0648 - affiliation: 1 -affiliations: - - name: North Carolina State University, Raleigh NC, USA - index: 1 -date: 15 April 2024 -bibliography: paper.bib ---- - -# Summary - -[MultiPrecisionArrays.jl](https://github.com/ctkelley/MultiPrecisionArrays.jl), -[@kelley:2024a; @kelley:2024b] provides data structures and -solvers for several variations of iterative refinement (IR). IR can speed up an LU matrix factorization for solving linear systems of equations by -factoring a low precision copy of the matrix and using that -low precision factorization in a iteration loop to solve the system. -For example, if high precision is double and low precision is single, -then the factorization time is cut in half. -The additional storage cost is the low precision copy, -so IR is at time vs storage trade off. IR has a long history -and a good account of the classical theory is in [@higham:1996]. - -# Statement of need - -The solution of linear systems of equations is a ubiquitous task -in computational science and engineering. A common method for dense -systems is Gaussian elimination done via an LU factorization, -[@higham:1996]. Iterative refinement is a way to reduce the factorization -time at the cost of additional storage. -[MultiPrecisionArrays.jl](https://github.com/ctkelley/MultiPrecisionArrays.jl) -enables IR with a simple interface in Julia [@Juliasirev:17] with an -IR factorization object that one uses in the same way as the one for LU. -The package offers several variants of IR, both classical -[@Wilkinson:48; @higham:1996], and some from the recent literature -[@CarsonHigham:2017; @amestoy:2024]. - -# Algorithm - -This package will make solving dense systems of linear equations faster by using the LU factorization and IR. While other factorizations can be used in IR, the -package is limited to LU for now. A very generic description of this -for solving a linear system $A x = b$ in a high (working) precision is - -__IR(A, b)__ - -- $x = 0$ - -- $r = b$ - -- Factor $A = LU$ in a lower precision - -- While $\| r \|$ is too large - - - $d = (LU)^{-1} r$ - - - $x = x + d$ - - - $r = b - Ax$ - -- end - -- end - - -In Julia, a code to do this would solve the linear system $A x = b$ in -in the working precision, say double, -by using a factorization in a lower (factorization) -precision, say single, within a residual correction iteration. -This means that one would need to allocate storage for a copy of $A$ -in the factorization precision and factor that copy. - -The multiprecision factorization ```mplu``` makes the low precision copy -of the matrix, factors that copy, and allocates some storage for the -iteration. The original matrix and the low precision factorization -are stored in a factorization object that you can use with ```\```. - -IR is a perfect example of a storage/time trade off. -To solve a linear system $A x = b$ in $R^N$ with IR, -one incurs the storage penalty of making a low -precision copy of $A$ and reaps the benefit of only having to -factor the low precision copy. - -# Installation - -The standard way to install a package is to type ```import.Pkg; Pkg.add("MultiPrecisionArrays")``` at the Julia prompt. One can run the unit tests with ```Pkg.test("MultiPrecisionArrays")```. -After installation, type ```using MultiPrecisionArrays``` when you want to use the functions in the package. - -There are only two direct dependencies outside of the Julia standard libraries. -The factorization in half precision (Float16) uses -[OhMyThreads.jl](https://github.com/JuliaFolds2/OhMyThreads.jl). The -GMRES and Bi-CGSTAB solvers for Krylov-IR methods are taken from -[SIAMFANL.jl](https://github.com/ctkelley/SIAMFANLEquations.jl) -[@kelley:2022b]. - -# Example - -Here is a simple example to show how ```mplu``` works. -We will follow that with some benchmarking on the cost of factorizations. -The computations were done with Julia 1.10.4 -on an Apple Mac Mini with an M2 pro processor and 32GB RAM. We used -OPENBLAS for LAPACK and the BLAS for this example. Other choices, such as the -[AppleAccelerate](https://github.com/JuliaLinearAlgebra/AppleAccelerate.jl) -framework would work equally well. - -In this example high (working) -precision is double, ```Float64```, and low (factorization) -precision is single, ```Float32```. -The matrix is the sum of the identity and a constant multiple of the trapezoid rule discretization of the Greens operator for $-d^2/dx^2$ on $[0,1]$ - -$$ -G u(x) = \int_0^1 g(x,y) u(y) \, dy -$$ - -where - -$$ -g(x,y) = \min(x,y) ( 1 - \max(x,y) ). -$$ - -The discretization for an $N$-point discretization is the $N \times N$ -maatrix -$$ -G_{ij} = g(x_i, x_j) /(N+1) -$$ -where $x_i = i/(N+1)$. - - -The code for construction $G$ is in the __/src/Examples__ directory. The file is __Gmat.jl__. You need to do -``` -using MultiPrecisionArrays -using MultiPrecisionArrays.Examples -``` -to use ```mplu``` and ```Gmat```. - -Now we will see how the results look. In this example we compare the result with iterative refinement with ```A\b```, which is LAPACK's LU. -As you can see the results are equally good. -Note that a factorization object ```MPF``` is the -output of ```mplu```. This is analogous to ```AF=lu(A)``` in LAPACK. - -``` -julia> using MultiPrecisionArrays - -julia> using MultiPrecisionArrays.Examples - -julia> using BenchmarkTools - -julia> N=4096; - -julia> # Build G on an N x N grid - -julia> G=Gmat(N); - -julia> # The operator is I - G - -julia> A = I - G; - -julia> # Create a test problem where the solution is x = ones(N) - -julia> x=ones(N); b=A*x; - -julia> # Compute the mplu and lu factorizations of A - -julia> MPF=mplu(A); AF=lu(A); - -julia> # Solve A x = b with both factorizations - -julia> z=MPF\b; w=AF\b; - -julia> # Compute relative errors and residuals for mplu - -julia> mpluerr=norm(z-x,Inf); mpluresid=norm(b-A*z,Inf)/norm(b,Inf); - -julia> # Compute relative errors and residuals for lu - -julia> luerr=norm(w-x,Inf); luresid=norm(b-A*w,Inf)/norm(b,Inf); - -julia> # Print the results - -julia> println("Errors: $mpluerr, $luerr. Residuals: $mpluresid, $luresid") -Errors: 4.44089e-16, 6.68354e-14. Residuals: 4.44089e-16, 6.68354e-14 -``` - -So the results are equally good. - -The compute time for ```mplu``` should be roughly half that of ```lu```. -A fair comparison is with ```lu!```, which does not allocate new storage -for the factorization. ```mplu``` factors a low precision array, -so the factorization cost is cut in half. Memory is a different story because -neither ```mplu``` nor ```lu!``` -allocate storage for a new high precision array, -but mplu allocates for a low precision copy, -so the memory and allocation cost for mplu is -50\% more than lu. One issue with smaller problems is that the triangular solve -does not parallelize as well as the factorization, so does not exploit -multi-core processor as well. We can see this in the IR solver times because -each iteration of IR needs a matrix-vector multiply and a triangular solve. - -``` -julia> using BenchmarkTools - -julia> @belapsed mplu($A) -8.60945e-02 - -julia> @belapsed lu!(AC) setup=(AC=copy($A)) -1.42840e-01 - -# And now for the solve times - -julia> @belapsed ldiv!($AF,bb) setup=(bb = copy($b)) -4.79117e-03 - -julia> @belapsed $MPF\$b -2.01195e-02 -``` - -So the total solve time (factorization and solve) is less for IR -by roughly 40%. - -# A Few Subtleties - -Within the algorithm one has to determine what the line -$d = (LU)^{-1} r$ means. Does one cast $r$ into the lower precision before -the solve or not? If one casts $r$ into the lower precision, then the solve -is done entirely in the factorization precision. If, however, $r$ remains -in the working precision, then the LU factors are promoted to the working -precision on the fly. This makes little difference if TW is double and -TF is single and there is a modest performance benefit to downcasting $r$ into -single. Therefore that is the default behavior in that case. -If TF is half precision, ```Float16```, then it is best to do the interprecision -transfers on the fly and if -one is using one of the Krylov-IR algorithms -[@amestoy:2024] then one must do the interprecision transfers on the fly -and not downcast $r$. - -There are two half precision (16 bit) formats. Julia has native support for IEEE 16 bit floats (Float16). A second format (BFloat16) has a larger exponent field and a smaller significand (mantissa), thereby trading precision for range. In fact, the exponent field in BFloat is the same size (8 bits) as that for single precision (Float32). The significand, however, is only 8 bits. Compare this to the size of the exponent fields for Float16 (11 bits) and single (24 bits). The size of the significand means that you can get in real trouble with half precision in either format and that IR is more likely to fail to converge. GMRES-IR -can mitigate the convergence problems [@amestoy:2024] by using the -low-precision solve as a preconditioner. We support both GMRES -[@saad:1986] and BiCGSTAB [@VanderVorst:1992] as solvers for -Krylov-IR methods. One should also know that LAPACK and the BLAS do not yet -support half precision arrays, so working in Float16 will be slower than -using Float64. - -The classic algorithm from [@Wilkinson:48] and its recent extension -[@CarsonHigham:2017] evaluate the residual in a higher precision -that the working precision. This can give improved accuracy for -ill-conditioned problems at a cost of the interprecision transfers -in the residual computation. This needs to be implemented with some -care and [@demmel:06] has an excellent account of the details. - -__MultiPrecisionArrays.jl__ provides -infrastructure to manage these things and we refer the reader to -[@kelley:2024b] for the details. - -# Projects using __MultiPrecisionArrays.jl__. - -This package was motivated by -the use of low-precision factorizations in Newton's method -[@kelley:2022a, @kelley:2022b] and the interface between -a preliminary version of this -package and the solvers from [@kelley:2022c] was reported in -[@kelley:2023]. That paper used a three -precision form of IR (TF=half, TW=single, nonlinear residual -computed in double) and required direct use of multiprecision -constructors that we do not export in __MultiPrecisionArrays.jl__. -We will fully support the application to nonlinear solvers in -a future version. We give a detailed account of interprecision -transfers in [@kelley:2024c] and use -__MultiPrecisionArrays.jl__ to generate the table in that paper. - - -# Other Julia Packages for IR - -The package -[IterativeRefinement.jl](https://github.com/RalphAS/IterativeRefinement.jl) -is an implementation of the IR method from -[@dongarra:1983]. It has not been updated in four years. - -The unregistered -package [Itref.jl](https://github.com/bvieuble/Itref.jl) implements -IR and the GMRES-IR method from [@amestoy:2024] and was used to obtain -the numerical results in that paper. It does not provide the -data structures for preallocation that we do and does not seem to have -been updated lately. - - -# Acknowledgements - -This work was partially supported by -Department of Energy grant DE-NA003967. -Any opinions, findings, and conclusions or recommendations -expressed in this material are those of the author and -do not necessarily reflect the views of the United States Department of Energy. - - -# References diff --git a/Joss_Paper/paper.pdf b/Joss_Paper/paper.pdf index 9b16b8eeb1a1d9e0836f11b6babe0de9b59b01a0..e25c344f14ce40cd979c0a01212fb45acf3c91d8 100644 GIT binary patch delta 61478 zcmV)JK)b)L+zy4u4Y2kK0X3JAbpt1p9t#?O%Z?oKhUz?4`47r&-Anr-yAQH zgW=N#P7b>XoJ)Q`)PoYGdUSR5WRt_fJMJo#L{b!AB2~pF4*&h@Z|R3mfBx}Fqzf?# zE0_{`l0q;Oz~q*IKOy`RLGb$N>4&ROfLwcj z=7kT_r{7lz`t)`B^_g>NI9CdOCHy(KHZQ+@`ssWoh}ikZg^`VBt(dfa zGR8A2ec?$6LF?3oT&73D*mPq8a~7L__|sQ0UH%Cew{Y%hX>@tUwVh7RaV7^&Iwpky zqB>4m=B=J=fJJela`K&XV>#Cs_yw+i>N9Xtz%_7J$9A3D4E}QXy=KnGKB=F4kse@~ 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provides data structures and solvers for several variations of iterative refinement (IR). IR can speed up an LU matrix factorization for solving linear systems of equations by factoring a low precision copy of the matrix and using that -low precision factorization in a iteration loop to solve the system. +low precision factorization in an iteration loop to solve the system. For example, if high precision is double and low precision is single, then the factorization time is cut in half. The additional storage cost is the low precision copy, @@ -70,7 +70,7 @@ __IR(A, b)__ - end -In Julia, a code to do this would solve the linear system $A x = b$ in +In Julia, a code to do this would solve the linear system $A x = b$ in the working precision, say double, by using a factorization in a lower (factorization) precision, say single, within a residual correction iteration. From fbb3bce312c26769e022b3ac0ac5c07c8d791a98 Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Wed, 25 Sep 2024 11:16:18 -0400 Subject: [PATCH 12/23] update bib --- Joss_Paper/paper.bib | 6 ++++-- 1 file changed, 4 insertions(+), 2 deletions(-) diff --git a/Joss_Paper/paper.bib b/Joss_Paper/paper.bib index c25a1767..e9b1e04c 100644 --- a/Joss_Paper/paper.bib +++ b/Joss_Paper/paper.bib @@ -38,7 +38,8 @@ @article{dongarra:1983 journal="SIAM Journal on Numerical Analysis", volume=20, pages="23--45", -year=1983 +year=1983, +doi={10.1137/0720002} } @techreport{Wilkinson:48, @@ -155,7 +156,8 @@ @article{VanderVorst:1992 journal = {SIAM J. Sci. Stat. Comp.}, year=1992, pages="631--644", -volume=13 +volume=13, +doi={10.1137/0913035} } @ARTICLE{saad:1986, From bcc3be48dd751fe7604dccb1ec4b73a22d5b97ce Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Wed, 25 Sep 2024 11:26:57 -0400 Subject: [PATCH 13/23] reword "iteration loop" --- Joss_Paper/paper.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Joss_Paper/paper.md b/Joss_Paper/paper.md index 1e21803f..aa7d1286 100644 --- a/Joss_Paper/paper.md +++ b/Joss_Paper/paper.md @@ -22,7 +22,7 @@ bibliography: paper.bib [@kelley:2024a; @kelley:2024b] provides data structures and solvers for several variations of iterative refinement (IR). IR can speed up an LU matrix factorization for solving linear systems of equations by factoring a low precision copy of the matrix and using that -low precision factorization in an iteration loop to solve the system. +low precision factorization in an iteration to solve the system. For example, if high precision is double and low precision is single, then the factorization time is cut in half. 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zV49!IU#t4GAzxd@=ezzrx#h*n*;1iTG6v4xi7jsJxgiesKK6{}*{P~O(lzfS(15_I zva|o3-Jl9HxLz6Af{E=WIE zlTxI7oMYTn_qUJ?aRhy{cqF2bhmN~l~AJg3HdF%L`vEi`J z(Ku;eA1-~KoLDcea7|13YU6@W-x?b(eSSuDNV_nV>KKK2m%j8>Rb=7x7a1X!<5>$) z;phSJdEb7W_oVsr@r;~a@lWcWg`r#TUGAk-bTsb)h;r|J%p^4*I@lYcB146S9M@ZR zp&~=%_M+R(Es6ZET1&96?>cc&*Sl{j9}k*AgR$1q&#s(38f<*@s!was@kDlSYt-3? zDv6wt))@Ip>FlQmy(n)ico#Jel%`v*v$R%T9;etlg<0L-A<%4O>abq0GCVgSXY2Zz z7VFw@B(|8Rv~z!7xU%!?m;#b@s1|uV%vVvzR{&S*XjTj_r1>Ty8RiK7{OKQm5gnJj7lLp-&@v`D&boL4vuvpfV-Lpr@yF z*0!?dl8&XAz+g=H$Na1*sHT*68SnVb>&cpzr-{b&(#D%{X3v>hHAaZxq4+^TKF{Y-aI48BN0Oy#uAzr@%4O^WP%zlUZ;F^%JET6ff!fF z?A%}~H#=c(&eYe=6|LewhN&Gq9~iT6Bj0xgE`~jXg6u&F7We?pA|<6|pw06C04mSZ ATL1t6 From c2e050c13a35117d96a5440e0dca0c82069bad0a Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Thu, 26 Sep 2024 13:00:06 -0400 Subject: [PATCH 15/23] bib fixes --- Joss_Paper/paper.bib | 11 ++++++----- Joss_Paper/paper.md | 2 +- 2 files changed, 7 insertions(+), 6 deletions(-) diff --git a/Joss_Paper/paper.bib b/Joss_Paper/paper.bib index e9b1e04c..39794ad1 100644 --- a/Joss_Paper/paper.bib +++ b/Joss_Paper/paper.bib @@ -79,12 +79,13 @@ @book{kelley:2022b author="C. T. Kelley", title="{Solving Nonlinear Equations with Iterative Methods: Solvers and Examples in Julia}", -year=2022, -publisher="SIAM", +publisher="SIAM, Philadelphia", address="Philadelphia", -isbn="978-1-611977-26-4", -series="Fundamentals of Algorithms", -doi={10.1137/1.9781611977271}, +Published:2022 +ISBN:978-1-61197-726-4 +eISBN:978-1-61197-727-1 +https://doi.org/10.1137/1.9781611977271 +Book Series Name:Fundamentals of Algorithms number=20 } diff --git a/Joss_Paper/paper.md b/Joss_Paper/paper.md index aa7d1286..f12d831c 100644 --- a/Joss_Paper/paper.md +++ b/Joss_Paper/paper.md @@ -19,7 +19,7 @@ bibliography: paper.bib # Summary [MultiPrecisionArrays.jl](https://github.com/ctkelley/MultiPrecisionArrays.jl), -[@kelley:2024a; @kelley:2024b] provides data structures and +[@kelley:2024b] provides data structures and solvers for several variations of iterative refinement (IR). IR can speed up an LU matrix factorization for solving linear systems of equations by factoring a low precision copy of the matrix and using that low precision factorization in an iteration to solve the system. From a64cb9f0975e731521a071de9d1f16b6e15ab4d4 Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Thu, 26 Sep 2024 13:03:08 -0400 Subject: [PATCH 16/23] bib fix --- Joss_Paper/paper.bib | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/Joss_Paper/paper.bib b/Joss_Paper/paper.bib index 39794ad1..8fe50bb0 100644 --- a/Joss_Paper/paper.bib +++ b/Joss_Paper/paper.bib @@ -81,11 +81,11 @@ @book{kelley:2022b Solvers and Examples in Julia}", publisher="SIAM, Philadelphia", address="Philadelphia", -Published:2022 -ISBN:978-1-61197-726-4 -eISBN:978-1-61197-727-1 -https://doi.org/10.1137/1.9781611977271 -Book Series Name:Fundamentals of Algorithms +Published:2022, +ISBN:978-1-61197-726-4, +eISBN:978-1-61197-727-1, +https://doi.org/10.1137/1.9781611977271, +Book Series Name:Fundamentals of Algorithms, number=20 } From a309c5a39d452bb74493cc099238ec15db068679 Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Thu, 26 Sep 2024 13:07:04 -0400 Subject: [PATCH 17/23] fix bib bugs --- Joss_Paper/paper.bib | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/Joss_Paper/paper.bib b/Joss_Paper/paper.bib index 8fe50bb0..95905497 100644 --- a/Joss_Paper/paper.bib +++ b/Joss_Paper/paper.bib @@ -81,11 +81,11 @@ @book{kelley:2022b Solvers and Examples in Julia}", publisher="SIAM, Philadelphia", address="Philadelphia", -Published:2022, -ISBN:978-1-61197-726-4, -eISBN:978-1-61197-727-1, -https://doi.org/10.1137/1.9781611977271, -Book Series Name:Fundamentals of Algorithms, +year=2022, +ISBN ="978-1-61197-726-4", +eISBN = "978-1-61197-727-1", +doi="10.1137/1.9781611977271", +Book Series Name="Fundamentals of Algorithms", number=20 } From 063b1324a82cd98e2f01d4485e26b84c7100757d Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Thu, 26 Sep 2024 13:10:15 -0400 Subject: [PATCH 18/23] bib fixes --- Joss_Paper/paper.bib | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Joss_Paper/paper.bib b/Joss_Paper/paper.bib index 95905497..4e9c5db7 100644 --- a/Joss_Paper/paper.bib +++ b/Joss_Paper/paper.bib @@ -85,7 +85,7 @@ @book{kelley:2022b ISBN ="978-1-61197-726-4", eISBN = "978-1-61197-727-1", doi="10.1137/1.9781611977271", -Book Series Name="Fundamentals of Algorithms", +series="Fundamentals of Algorithms", number=20 } From 227fc4563ea15a3eba7e89338fbd4a5d67cfb741 Mon Sep 17 00:00:00 2001 From: Tim Kelley Date: Thu, 26 Sep 2024 13:17:32 -0400 Subject: [PATCH 19/23] updates --- Joss_Paper/paper.md | 2 +- Joss_Paper/paper.pdf | Bin 222633 -> 222251 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/Joss_Paper/paper.md b/Joss_Paper/paper.md index f12d831c..038df48e 100644 --- a/Joss_Paper/paper.md +++ b/Joss_Paper/paper.md @@ -141,7 +141,7 @@ This package was motivated by the use of low-precision factorizations in Newton's method [@kelley:2022a, @kelley:2022b] and the interface between a preliminary version of this -package and the solvers from [@kelley:2022c] was reported in +package and the solvers from [@kelley:2022b, @kelley:2022c] was reported in [@kelley:2023]. 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Joss_Paper/paper.pdf | Bin 222251 -> 222255 bytes 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/Joss_Paper/paper.md b/Joss_Paper/paper.md index 038df48e..fe08014e 100644 --- a/Joss_Paper/paper.md +++ b/Joss_Paper/paper.md @@ -139,9 +139,9 @@ infrastructure to manage these things and we refer the reader to This package was motivated by the use of low-precision factorizations in Newton's method -[@kelley:2022a, @kelley:2022b] and the interface between +[@kelley:2022a; @kelley:2022b] and the interface between a preliminary version of this -package and the solvers from [@kelley:2022b, @kelley:2022c] was reported in +package and the solvers from [@kelley:2022b; @kelley:2022c] was reported in [@kelley:2023]. 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a/Joss_Paper/paper.bib b/Joss_Paper/paper.bib index 4e9c5db7..f91a61b7 100644 --- a/Joss_Paper/paper.bib +++ b/Joss_Paper/paper.bib @@ -151,7 +151,7 @@ @article{demmel:06 } @article{VanderVorst:1992, -author="H. A. van der Vorst", +author="H. A. {van der Vorst}", title="{Bi-CGSTAB}: A fast and smoothly converging variant to {Bi-CG} for the solution of nonsymmetric systems", journal = {SIAM J. Sci. Stat. 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